Method of manufacturing sheet, device and program for controlling sheet thickness, and sheet

ABSTRACT

A method of controlling the thickness of sheets manufactured by the extrusion of a material from a die having a plurality of thickness adjusting means by repeating, at specified timings, the following steps of 1) measuring the distribution of thickness of the sheets in lateral direction, 2) evaluating a predicted future variation in sheet thickness by using a specified evaluation function and based on a process model representing a relation between the amount of operation and the sheet thickness and sheet thickness measured values and leading an operating amount time series to minimize the evaluation function, and 3) outputting at least the initial amount of operation of the led operating amount time series to the thickness adjusting means.

TECHNICAL FIELD

The present invention relates to a method of manufacturing a sheet suchas a film, a device for controlling thickness of a sheet, a program forcontrolling thickness of a sheet, and a sheet.

BACKGROUND ART

A conventional sheet production process, in which the thickness of asheet such as a macromolecular film is controlled in the transversedirection to have a desired profile such as a uniform thickness, isdescribed below in referent to FIGS. 2 and 3.

A macromolecular polymer as a raw material is extruded from an extruder3 while being widened in the transverse direction perpendicular to thepaper surface of FIG. 2 using a die 4, to form a sheet 1, and the sheet1 is stretched by a stretching machine 2 in the machine direction (sheetrunning direction) and the transverse direction (sheet transversedirection), and sheet 1 is wound by a winder 6. The die 4 has pluralthickness adjusting means 10 provided at equal intervals in thetransverse direction. The thickness adjusting means, for example,heaters or gap adjusters, have function of changing the amounts of thedischarged polymer. A thickness gauge 8 measures the thicknessdistribution of the sheet 1 in the transverse direction of the sheet,and a control means 9 manipulates plural thickness adjusting means 10based on the measured values at the positions corresponding to therespective thickness adjusting means.

Widely used control means is composed of control loops independentlyprovided for the respective thickness adjusting means, and for each ofthe control loops, this control means carries out known PID control, inwhich the result of the proportional-plus-integral-plus-derivativecomputation of the deviation between a measured thickness value and atarget value is delivered as a manipulated variable to each of thethickness adjusting means. Japanese Patent No. 3,021,135 discloses athickness controller using modern control theory as the thicknesscontrol means.

The above-mentioned conventional control system, in which independentcontrol loops are provided for the respective thickness adjusting means,cannot perform sufficiently satisfactory control yet. One of the reasonsis that if one of the thickness adjusting means is manipulated, aninterference phenomenon occurs in which the sheet thickness vary also atthe positions corresponding to the adjacent thickness adjusting means.For this reason, the control loops corresponding to the respectivethickness adjusting means interfere with each other, and even if themanipulated variables are computed based on the deviations between thethickness values and the target values of the corresponding positions,for control, it can happen that thickness distribution dose not approachthe target values affected by the adjacent adjusting means or that thespeed of approaching the target values is very slow.

As another reason, there is a time lag after one of the thicknessadjusting means is manipulated till the result is reflected in thethickness result at the corresponding position, that is, a delay timeso-called in control occurs. So, if the gain of PID control is madelarger, too much manipulation is carried out before the result of themanipulated variable delivered to the thickness adjusting means isreflected in the thickness result at the corresponding position, and thecontrol becomes unstable. Therefore, the gain of control must be keptsmall to stabilize the control, and the control system is poor in quickresponsiveness.

Meanwhile, in the case where, for example, a polyester film is wound asa roll, it can happen that the wound roll is wrinkled or streaked ordisfigured at end faces, to extremely lower the value of the producedroll, or even to completely loose the commercial value.

To avoid this problem, it is proposed to improve the surface propertiesof the film, or to decrease the thickness irregularity, or to dispersethe thickness irregularity in the transverse direction of the film byoscillation.

However, the above-mentioned prior art have such a problem that theproperties of the film have to be changed, or that the productivity mustbe lowered, or that the improvement is insufficient. Especially when itis attempted to obtain a thinner film, these problems become moresignificant.

Furthermore, in recent years, the requirement for better roll formbecomes more intensive, and oscillation cannot solve the problem anymore.

The invention has been completed to solve these problems. Objects of theinvention are to provide a sheet thickness controller that can uniformlyand stably control the thickness of a sheet in the transverse directionover the entire width, and to provide a process for producing such asheet.

Another object of the invention is to provide a roll which form has lesswrinkles and streaks, with the productivity sustained at a high levelwithout changing the properties of the sheet.

DISCLOSURE OF THE INVENTION

The invention provides a method of manufacturing sheet, in which a rawmaterial is extruded and molded into a sheet using a die with pluralthickness adjusting means and the thickness of said sheet is controlledby the manipulated variables applied to said thickness adjusting means,characterized by repeating, at predetermined intervals, a step ofmeasuring the thickness distribution of the sheet in the transversedirection, a step of deriving manipulated variable time series in whicha predetermined evaluation function for evaluating the future sheetthickness changes predicted based on said measured values and on aprocess model expressing the relation between said manipulated variablesand sheet thickness values becomes a minimum value, and a step ofdelivering at least the first manipulated variables of the derivedmanipulated variable time series to said thickness adjusting means.

The invention provides also a device for controlling sheet thickness, inwhich manipulated variables are applied to sheet thickness adjustingmeans at corresponding positions based on the measured sheet thicknessvalues at respective positions of a sheet in the transverse directionmeasured by a thickness measuring means for measuring the thicknessdistribution of the sheet in the transverse direction; comprising amanipulated variable time series deriving means for deriving manipulatedvariable time series in which a predetermined evaluation function forevaluating the future sheet thickness changes predicted based on saidmeasured values and on a process model expressing the relation betweensaid manipulated variables and sheet thickness values becomes a minimumvalue and a manipulated variable delivering means for delivering atleast the first manipulated variables of the derived manipulatedvariable time series to said thickness adjusting means.

Moreover, the invention provides a program, for letting a computerperform the action of repeating, at predetermined intervals, a step ofentering the measured thickness values at the respective positions of asheet in the transverse direction, a step of computing the differencesbetween the target thickness values and the measured thickness values atthe respective positions, and a step of computing the manipulatedvariables to be applied to thickness adjusting means based on thedifferences at the respective positions, characterized in that the stepof computing manipulated variables includes a step of derivingmanipulated variable time series in which a predetermined evaluationfunction for evaluating the future sheet thickness changes predictedbased on said measured values and on a process model expressing therelation between said manipulated variables and sheet thickness valuesbecomes a minimum value and a step of delivering at least the firstmanipulated variables of the derived manipulated variable time series tosaid thickness adjusting means.

Still furthermore, the invention provides a sheet, obtained by extrudingand molding a raw material using a die with plural thickness adjustingmeans, characterized in that the power spectrum of the thickness profileof the sheet in the transverse direction expressed by the followingformula P = F(ω)(F(ω))^(*) F(ω) = ∫_(−∞)^(∞)f(x)𝕖^(−jω  x)𝕕x(where f(x) is the thickness profile of the sheet in the transversedirection (in μm), F(ω) is the Fourier transform of f(x), x is aposition in the transverse direction of the sheet (in m), ω is a wavenumber (in m⁻¹), and F(ω)* is the conjugate complex number of F(ω); andj is an imaginary number, and j²=−1) and the average sheet thickness T(μm) satisfy the following relation: The mean value X1 of the powers ofsmaller than a predetermined wave number a is 0.2×T² or less and issmaller than the mean value X2 of the powers of wave number a andlarger.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a sheet thickness control flowchart in an embodiment of theinvention.

FIG. 2 is a schematic view of system of sheet production equipment in anembodiment of the invention.

FIG. 3 is an enlarged perspective view of a key portion of the die shownin FIG. 2.

FIG. 4 is a diagram showing the basic constitution of thickness controlin an embodiment of the invention.

FIG. 5 is a flowchart showing a control means design method in anembodiment of the invention.

FIG. 6 is a drawing showing the relation between thickness adjustingmeans and sheet thickness measurement in an embodiment of the invention.

FIG. 7 is a diagram showing a pattern of manipulated variables in anembodiment of the invention.

FIG. 8 is a diagram showing the details of a control means in anembodiment of the invention.

FIG. 9 is a flowchart for leveling the manipulation variables in anembodiment of the invention.

FIG. 10 is a flowchart for leveling the manipulation variables in anembodiment of the invention.

FIG. 11 is a block diagram showing a sheet thickness control process inan example of the invention.

FIG. 12 is a diagram showing the change of sheet thickness with thelapse of time in the case where a heat bolts is alternately pushed orpulled, in an embodiment of the invention.

FIG. 13 is a diagram showing the change of sheet thickness with thelapse of time in the case where the heat quantity is large or small, inan embodiment of the invention.

FIG. 14 is a block diagram showing a sheet thickness control process inan embodiment of the invention.

FIG. 15 is a side view of a produced roll having a winding mound.

FIG. 16 shows a result of sheet thickness control in the sheetproduction in a conventional process.

FIG. 17 shows a result of thickness control in the sheet production inan example of the invention.

FIG. 18 is a diagram showing a film thickness distribution in the casewhere the film thickness is controlled using the process of theinvention.

FIG. 19 is a diagram showing a film thickness distribution in the casewhere the film thickness is controlled using a conventional process.

FIG. 20 is a diagram showing a pattern of manipulated variables in anembodiment of the invention.

FIG. 21 is a diagram showing a pattern of differential values of therespectively adjacent manipulated variables in FIG. 20.

FIG. 22 is a graph showing a result of thickness control used in aproduction of sheet in an embodiment of the invention.

FIG. 23 is a graph showing a result of thickness control used in aproduction of sheet in the case where manipulated variables are leveledby a conventional method.

FIG. 24 is a diagram showing a pattern of manipulated variables in anembodiment of the invention.

FIG. 25 is a diagram showing a manipulation result in an embodiment ofthe invention.

FIG. 26 shows the sheet thickness pattern corresponding to positionsshown in FIG. 6.

FIG. 27 shows the sheet thickness pattern corresponding to positionsshown in FIG. 7.

FIG. 28 is a graph showing a result of thickness control in sheetproduction in the case where manipulated variables are leveled by aconventional method.

FIG. 29 is a diagram showing the change of film thickness with the lapseof time in the case where heat bolts are heated in an embodiment of theinvention.

FIG. 30 is a diagram showing the change of film thickness with the lapseof time in the case where heat bolts are cooled in the example shown inFIG. 29.

FIG. 31 is a graph showing an outer diameter profile of a produced rollin the examples shown in FIGS. 29 and 30.

FIG. 32 is a diagram showing the power spectrum of the sheet thicknessprofile in the transverse direction in an embodiment of the invention.

MEANINGS OF SYMBOLS

-   1: Sheet-   2: Stretching machine-   3: Extruder-   4: Die-   5: Cooling roll-   6: Winder-   7: Carrier roll-   8: Thickness meter-   9: Control means-   10: Thickness adjusting means-   11: Slit-   21: Manipulated variable computing means-   22: Manipulated variable delivering means-   23: Manipulated variables-   24: Measured thickness values-   25: Deviation data-   26: Sheet production process-   27: Manipulated variables to be corrected-   28: Manipulated variable computing means-   211: Basic heat quantity computing means-   212: Control heat quantity computing means-   213: Control differential value computing means-   214: Heat quantity computing means-   30: Change of sheet thickness with the lapse of time in the case    where a heat bolt is pulled-   31: Change of sheet thickness with the lapse of time in the case    where a heat bolt is pushed-   32: Sheet thickness variation after lapse of certain time in the    case where a heat bolt is pulled-   33: Sheet thickness variation after lapse of certain time in the    case where a heat bolt is pushed-   34: Certain period of time after applying a heat quantity-   35: Change of sheet thickness with the lapse of time in the case    where the heat quantity is large-   36: Change of sheet thickness with the lapse of time in the case    where the heat quantity is small-   37: Change of sheet thickness with the lapse of time in the case    where the time constant is smaller than that of 36-   40: Second desired value correcting means-   41: Integrating means-   42: First desired values-   43: Second desired values-   44: Deviations-   50: Produced roll-   51: Paper tube-   52: Winding mound-   60: Manipulated variable to be corrected-   71: Change of film thickness with the lapse of time in the case    where heat bolts are pushed-   72: Functional approximation curve of 71-   73: Change of film thickness with the lapse of time in the case    where heat bolts are pulled-   74: Functional approximating curve of 73-   75: 10% variation of film thickness-   76: 90% variation of film thickness-   81: Outer diameter profile of a produced roll in the case where the    film is produced by a conventional process-   82: Outer diameter profile of a produced roll in the case where the    film is produced by the process of the invention-   90: Power spectrum

THE BEST MODES FOR CARRYING OUT THE INVENTION

The invention is described below in reference to drawings.

An example of the sheet production process, in which the thickness of asheet such as a macromolecular film is controlled in the transversedirection of the sheet to have a desired profile, such as a uniformthickness, is described below in reference to FIGS. 2, 3 and 4.

FIG. 2 is a schematic view of general sheet production equipment, andFIG. 3 is an enlarged perspective view of the die shown in FIG. 2.

FIG. 4 is a block diagram showing a method of controlling the sheetthickness.

A polymer is extruded from an extruder 3 and widened in the transversedirection perpendicular to the paper surface of FIG. 2 by a die 4, toform a sheet, and the sheet is stretched in the machine direction and inthe transverse direction by a stretching machine 2, and wound by awinder 6.

The die 4 has plural thickness adjusting means 10 disposed at equalintervals in the transverse direction. The thickness adjusting means 10can be either of bolt method or heater method; according to the boltmethod, bolts as thickness adjusting means are disposed to change thegap 11 of the die 4 mechanically, thermally or electrically for changingthe amount of the discharged polymer, and according to the heatermethod, heaters as thickness adjusting means are disposed and changesthe generated heat for changing the viscosity, hence flow velocity ofthe polymer at the portion, for changing the discharged amount.Furthermore, the sheet production equipment is provided with a thicknessgauge 8 for measuring the thickness distribution of the sheet in thetransverse direction and a control means 9 for controlling the thicknessadjusting means 10 based on the thickness distribution.

The thickness gauge 8 measures the thickness values of the sheet as athickness distribution in the transverse direction of the sheet. Thethickness meter 8 can be any optional thickness measuring instrumentthat uses the absorptivity of β rays, infrared rays, ultraviolet rays, Xrays or the like, or the light interference phenomenon, etc.

The control means 9 receives the measured sheet thickness distributionvalues in the transverse direction measured by the thickness gauge 8,and obtains the measured sheet thickness values corresponding to therespective manipulating points of the thickness adjusting means 10 fromthem, and derives the manipulated variables for the respectivemanipulating points by computing the control actions based on themeasured sheet thickness values corresponding to the respectivemanipulating points as described later by using a manipulated variablecomputing means 21, and delivers the manipulated variables to therespective manipulating points at predetermined intervals by using amanipulated variable delivering means 22.

The manipulated variables for the respective manipulating points areapplied to the thickness adjusting means 10 through a power unit notillustrated. In the case of the heat bolt method for thermally expandingand contracting bolts, the power unit supplies electric powers to theheaters attached to the bolts, to heat the bolts which are expanded orcontracted depending on the powers, for adjusting the gap 11. Also inany other method, electric powers are supplied to actuate the thicknessadjusting means 10, and the actuated thickness adjusting means controlthe sheet to have a target profile.

The action of the control means 9 is described below in detail.

FIG. 1 is a flowchart of the control means 9. At every time point t(t=0, 1, 2, . . . ) after start of control, the sheet thicknessdistribution measured by the thickness meter 8 is applied to obtain themeasured sheet thickness values corresponding to the respectivemanipulating points of the thickness adjusting means 10, and in themanipulated variable time series deriving step described later,manipulated variable time series are derived. Then, in a manipulatedvariable delivering step, the manipulated variables to be actuallydelivered to the thickness adjusting means 10 are decided from thederived manipulated variable time series, and are delivered. Thisprocess is repeated till the control is completed.

The following description is made using discrete times. It is preferablethat each control time interval is the time taken for the thicknessgauge 8 to measure the thickness distribution of the sheet 1 in thetransverse direction of the sheet, or a multiple of the time.Ordinarily, the intervals are tens of seconds to several minutes. Thecontrol timings are not necessarily required to be fixed cycles, and canbe changed as required, depending on the stability condition of theprocess, etc. For example, in the beginning of production, the controlcan be carried out at short cycles, and during stable production, thecontrol can be carried out at long cycles.

In the case where a process model is used, when a certain manipulatedvariable time series is delivered, how the sheet thickness will changecan be predicted. In the manipulated variable time series deriving step,what manipulated variables should be delivered for control to optimizethe predicted sheet thickness, that is, to keep a predeterminedevaluation function at a minimum value.

The computation in the manipulated variable time series deriving step isto obtain manipulated variable time series from a predeterminedmanipulated variable time series deriving formula using the measuredsheet thickness values and the manipulated variables delivered tillthen. The concept for arriving at this manipulated variable timederiving formula is described below.

At first, considered is a process model expressing how the sheetthickness changes when manipulated variables are applied to thicknessadjusting means 10. This process model expresses, as a numericalformula, the lag in the action of the thickness adjusting means afterthe delivery of the manipulated variables, the delay time taken for thesheet 1 to be carried from the die 4 to the position of the thicknessmeter 8, and the delay time taken for the thickness meter 8 to measurethe thickness profile in the transverse direction, and also theinterference that when one thickness adjusting means is manipulated, thesheet changes in thickness at the positions corresponding to theadjacent thickness adjusting means. As far as the above conditions aresatisfied, any process model can be used. However, if an individualmodel is used for each thickness adjusting means, enormous time andlabor are needed, and the time series deriving formula becomes toocomplicated. So, it is preferable that the process model is expressedusing a product obtained by multiplying a scalar transfer functionexpressing the relation between the manipulated variables of thicknessadjusting means and the film thickness values at the correspondingpositions, by a constant matrix expressing the interferences betweenindividual thickness adjusting means in which at least the diagonalcomponent is not zero. With this, the manipulated variable time seriescomputation can be simplified. Such a process model can be expressed bythe following formula using, for example, a discrete time transferfunction. $\begin{matrix}{{Formula}\quad 1\text{:}} \\{{y_{M}(z)} = {\frac{{b_{q}z^{- q}} + {a_{q - 1}z^{- {({q - 1})}}} + \cdots + {b_{1}z^{- 1}}}{{a_{p}z^{- p}} + {a_{p - 1}z^{- {({p - 1})}}} + \cdots + 1}{{wu}(z)}}}\end{matrix}$

In the above formula, y_(M) and u respectively are the sheet thicknessvalues and manipulated variables at the measuring positionscorresponding to respective thickness adjusting means, being vectorshaving elements as many as the number N of thickness adjusting means 10;p and q are the degrees of the discrete time transfer function; and aand b are the respective coefficients; and p, q, a and b are decidedconsidering the delay time and lag in the actual sheet productionprocess.

W is an N×N matrix expressing the interferences between the individualthickness adjusting means, and expressed by the following formula.$\begin{matrix}{{Formula}\quad 2\text{:}} \\{W = \begin{bmatrix}1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \cdots & \quad & \cdots & \quad & 0 \\\alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & \quad & \quad & 0 \\\alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & \quad & 0 \\0 & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & \quad \\0 & 0 & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & ⋰ & \quad & \quad \\0 & \quad & ⋰ & ⋰ & ⋰ & 1 & ⋰ & \quad & \quad & \quad \\\vdots & \quad & \quad & ⋰ & \quad & \quad & ⋰ & \quad & ⋰ & \quad \\\quad & \quad & \quad & \quad & \quad & ⋰ & ⋰ & ⋰ & ⋰ & \alpha_{2} \\\quad & \quad & \quad & \quad & \quad & ⋰ & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} \\0 & \quad & \cdots & \quad & \cdots & \quad & 0 & \alpha_{2} & \alpha_{1} & 1\end{bmatrix}}\end{matrix}$

In the above formula, α₁ (≧0) is the rate at which the sheet thicknessvalues at the positions corresponding to one of the first adjacentthickness measuring means change, and α₂ (≧0) is the rate at which thesheet thickness values at the positions corresponding to one of thesecond adjacent thickness adjusting means change. They are calledinterference rates in this specification. When a certain thicknessadjusting means is manipulated, the sheet thickness at the sheetthickness measuring position corresponding to the thickness adjustingmeans changes, and the interference rates are the values expressing howmuch the sheet thickness changes at the measuring positionscorresponding to the adjacent thickness adjusting means in this case.That is, when a certain thickness adjusting means is manipulated, thesheet thickness changes not only at the manipulating position but alsoin a certain peripheral region due to the rigidity of the die and theinfluence in the stretching process.

In the above formula, the rate at which the sheet thickness values atthe positions corresponding to both the third and farther adjacentthickness adjusting means change is assumed to be 0, but α₃ (≧0) andmore may also be considered. However, it is preferable to assume that α₃(≧0) and rates at farther thickness adjusting means are 0, since thecomputation is simpler with little effect on the computation result.Furthermore, as described later, the values of α₁ and α₂ of respectiverows can also be different from row to row.

From the process model, ifB_(i)=b_(i)W,the sheet thickness y_(M) (t) at time point t can be expressed byFormula 7: $\begin{matrix}{{y_{M}(t)} = {{a_{1}{y_{M}\left( {t - 1} \right)}} + {a_{2}{y_{M}\left( {t - 2} \right)}} + \cdots + {a_{p}{y_{M}\left( {t - p} \right)}} +}} \\{{B_{1}{u\left( {t - 1} \right)}} + {B_{2}{u\left( {t - 2} \right)}} + \cdots + {B_{q}{u\left( {t - q} \right)}}}\end{matrix}$Furthermore, ifΔy _(M)(t)=y _(M)(t)−y _(M)(t−1), andΔu(t)=u(t)−u(t−1),the sheet thickness values y_(M) (t+1) and y_(M) (t+2) at future timepoints t+1 and t+2 can be expressed respectively by $\begin{matrix}{{Formula}\quad 8\text{:}} \\\begin{matrix}{{y_{M}\left( {t + 1} \right)} = {{y_{M}(t)} + {a_{1}\Delta\quad{y_{M}(t)}} + {a_{2}\Delta\quad{y_{M}\left( {t - 1} \right)}} + \cdots +}} \\{{a_{p}\Delta\quad{y_{M}\left( {t + 1 - p} \right)}} + {B_{1}\Delta\quad{u(t)}} + {B_{2}\Delta\quad{u\left( {t - 1} \right)}} + \cdots +} \\{B_{q}\Delta\quad{u\left( {t + 1 - q} \right)}}\end{matrix} \\\begin{matrix}{{y_{M}\left( {t + 2} \right)} = {{y_{M}(t)} + {\left( {{\left( {1 + a_{1}} \right)a_{1}} + a_{2}} \right)\Delta\quad{y_{M}(t)}} + \cdots +}} \\{{\left( {1 + a_{1}} \right)a_{p}\Delta\quad{y_{M}\left( {t + 1 - p} \right)}} + {B_{1}\Delta\quad{u\left( {t + 1} \right)}} +} \\{{\left( {{\left( {1 + a_{1}} \right)B_{1}} + B_{2}} \right)\Delta\quad{u(t)}} + \cdots + {\left( {1 + a_{1}} \right)B_{q}\Delta\quad{u\left( {t + 1 - q} \right)}}}\end{matrix}\end{matrix}$If this is applied recursively, the sheet thickness values y_(M) (t+j)at time point t+j (j>1) can expressed as follows, using sheet thicknessvalues y_(M)(t−1), . . . , y_(M)(t−p), and manipulated variables u(t−q),. . . , u(t+j−1). $\begin{matrix}{{Formula}\quad 9\text{:}} \\\begin{matrix}{{y_{M}\left( {t + j} \right)} = {{y_{M}(t)} + {h_{j,1}\Delta\quad{y_{M}(t)}} + {h_{j,2}\Delta\quad{y_{M}\left( {t - 1} \right)}} + \cdots +}} \\{{h_{j,p}\Delta\quad{y_{M}\left( {t + 1 - p} \right)}} + {g_{j,1}\Delta\quad{u\left( {t + j - 1} \right)}} +} \\{{g_{j,2}\Delta\quad{u\left( {t + j - 2} \right)}} + \cdots + {g_{j,q}\Delta\quad{u\left( {t + 1 - q} \right)}}}\end{matrix}\end{matrix}$

In the above formula, the sheet thickness values y_(M) (t−1), . . . ,y_(M) (t−p) and manipulated variables u (t−1), . . . , (t−p) are knownat time point t, and g and h can be obtained from the coefficients a andb of the transfer function shown in formula 1 and are known beforehandfrom the above process model. So, it can be said that the future sheetthickness values y_(M) (t+j) can be calculated if the manipulatedvariable time series u (t), . . . , u (t+j−1) delivered after the timepoint t are known.

The above sheet thickness values are obtained from a process model.However, the process model does not perfectly agree with the actualprocess, and the actual sheet thickness values become different becauseof various disturbances, etc. Therefore, even if the sheet thicknessvalues in the far future are obtained to derive the evaluation functionfor optical control, the manipulated variables are decided usinguncertain information with large errors after all unpreferably. So,considered are finite periods of time such as time m (an integer largerthan 0) taken to change the manipulated variables and the time P (aninteger larger than 0) taken to obtain the sheet thickness values. Thatis, in the case where the manipulated variables are changed from timepoint t to t+m−1 and kept constant thereafter, the sheet thicknessvalues of the period from time point (t+L) to (t+L+P−1) (L is aninteger) can be expressed by $\begin{matrix}\text{Formula~~10:} \\\begin{matrix}{\quad{\begin{bmatrix}{y_{M}\left( {t + L} \right)} \\{y_{M}\left( {t + L + 1} \right)} \\\vdots \\\vdots \\{y_{M}\left( {t + L + P - 1} \right)}\end{bmatrix} = {\begin{bmatrix}{y_{M}(t)} \\{y_{M}(t)} \\\vdots \\\vdots \\{y_{M}(t)}\end{bmatrix} +}}} \\{\begin{bmatrix}g_{L,L} & g_{L,{L - 1}} & \cdots & g_{L,1} & \cdots & 0 \\g_{{L + 1},{L + 1}} & g_{{L + 1},L} & \cdots & g_{{L + 1},1} & \cdots & 0 \\\vdots & \vdots & \quad & \vdots & \quad & \vdots \\\vdots & \vdots & \quad & \vdots & \quad & \vdots \\\vdots & \vdots & \quad & \vdots & \quad & \vdots\end{bmatrix}} \\{\left\lbrack \quad\begin{matrix}{\Delta\quad{u(t)}} \\{\Delta\quad{u\left( {t + 1} \right)}} \\\vdots \\\vdots \\{\Delta\quad{u\left( {t + m - 1} \right)}}\end{matrix} \right\rbrack +} \\{{\begin{bmatrix}g_{L,{L + 1}} & g_{L,{L + 2}} & \cdots \\g_{{L + 1},{L + 2}} & g_{{L + 1},{L + 3}} & \cdots \\g_{{L + 2},{L + 3}} & \vdots & \quad \\\vdots & \quad & \quad \\g_{{L + p - 1},{L + p}} & \quad & \quad\end{bmatrix}\begin{bmatrix}{\Delta\quad{u\left( {t - 1} \right)}} \\{\Delta\quad{u\left( {t - 2} \right)}} \\\vdots \\\vdots \\{\Delta\quad{u\left( {t - q + 1} \right)}}\end{bmatrix}} +} \\{\begin{bmatrix}h_{L,1} & h_{L,2} & \cdots \\h_{{L + 1},1} & h_{{L + 1},2} & \cdots \\\vdots & \vdots & \quad \\h_{{L + p - 1},1} & \cdots & \quad\end{bmatrix}\begin{bmatrix}{\Delta\quad{y(t)}} \\{\Delta\quad{y\left( {t - 1} \right)}} \\\vdots \\\vdots \\{\Delta\quad{y\left( {t - p + 1} \right)}}\end{bmatrix}}\end{matrix}\end{matrix}$The above can be expressed as a vector matrix as follows.Formula 11:y _(M) =y _(M0) +G _(F) Δu _(n) +G ₀ Δu ₀ +Q ₀ Δy _(M)

The above is the future sheet thickness values derived from the processmodel. On the other hand, at time point t, the thickness gauge 8measures the actual sheet thickness distribution, and from it, theactual sheet thickness values y (t) corresponding to the respectivemanipulating points of the thickness adjusting means 10 can be known.So, if they are used to predict the sheet thickness values at time pointt+J, the prediction formula y_(P) (t+j) is

Formula 12:y _(P)(t+j)=y _(M)(t+j)+y(t)−y _(M)(t)As described above, the formula for predicting the sheet thicknessvalues in the period from time point (t+L) to (t+L+P−1) is,Formula 13:y _(P) =y+G _(F) ΔU _(n) +G ₀ ΔU ₀ +Q ₀ Δy _(M)In the above formula, y is a vector as an array of M vectors y (t)respectively having elements as many as N. That is, y_(P) is time seriesexpressing the future sheet thickness changes predicted from the aboveformula.

Now, considered is the evaluation function for evaluating to ensure thatthe thickness prediction formula is optimized.

At first, from the sheet thickness profile y(t) measured as of timepoint t, set is a reference orbit Y_(R) (t+j) (j=1, 2, . . . ) arrivingat the desired thickness profile r (vector with N elements) at timepoint t+j.

The reference orbit can be set as required according to a conventionalmethod. It can be expressed by, for example, Formula 14:y _(R)(t+j)=β^(j−L+1) y(t)+(1−β^(j−L+1))rIf β is made closer to 0, an orbit for approaching the designed profiler faster is obtained. It is desirable that the deviation (quadratic formof it) between the sheet thickness prediction formula and this referenceorbit is smaller.

On the other hand, for the manipulated variables, it is desirable thatthe changes Δu of manipulated variables are smaller. Considering theabove points, as the evaluation function J,

Formula 15:J=(y _(R) −y _(P))^(T)Λ(y _(R) −y _(P))+ΔU _(n) ^(T) ΨΔu _(n)is used, and the manipulated variable time series in which this functionbecome a minimum value are derived. In the above formula,${{{{Formula}\quad 16}:y_{R}} = \begin{bmatrix}{y_{R}\left( {t + L} \right)} \\{y_{R}\left( {t + L + 1} \right)} \\\vdots \\\vdots \\{y_{R}\left( {t + L + P - 1} \right)}\end{bmatrix}},{\Lambda = \begin{bmatrix}\lambda_{1}^{2} & \quad & 0 \\\quad & ⋰ & \quad \\0 & \quad & \lambda^{2}\end{bmatrix}},{\Psi = \begin{bmatrix}\psi_{1}^{2} & \quad & 0 \\\quad & ⋰ & \quad \\0 & \quad & \psi_{P}^{2}\end{bmatrix}}$

In the above formula, the first term relates to the deviations betweenthe reference orbit and the predictive thickness till the targetthickness values are reached, and the second term relates to manipulatedvariables. Λ and Ψ decide the respective degrees of contribution.

In general, in the beginning of film production, since the deviationsfrom the desired thickness values are large, large manipulated variablesshould be applied to quickly make the deviations smaller, and duringstable film production, large manipulated variables should not beapplied since the deviations are small. So, it is preferable to prepareevaluation functions different in said relation between Λ and Ψ; toensure that the contribution of Ψ relating to the manipulated variablesis kept small in the beginning of film formation and that thecontribution of Ψ is kept large during stable film formation.

In this case, the necessary condition to ensure that the evaluationfunction J takes a minimum value is${{{Formula}\quad 17}:\frac{\partial J}{{\partial\Delta}\quad u_{n}}} = 0$and the Δu_(n) satisfying the above isFormula 18:Δu _(n)=(G _(F) ^(T) ΛG _(F)+Ψ)⁻¹ G _(F) ^(T)Λ(y _(R) −y−G ₀ Δu ₀ −Q ₀Δy _(M))and this is the manipulated variable time series deriving formula.

That is, in the manipulated variable time series deriving step, themeasured sheet thickness values y(t) obtained in the previous step aresubstituted into they and y_(R) of the above formula, and Δu₀ and Δy_(M)are updated based on the information of till time point t−1, to derivethe variations Δu_(n) of the manipulated variable time series, fordeciding u(t), . . . , u(t+m−1) from them.

Then in the manipulated variable delivering step, from the manipulatedvariable time series decided above, u(t) only is actually delivered tothe thickness adjusting means 10.

The manipulated variable time series deriving step and the manipulatedvariable delivering step are repeated at time points t, t+1, t+2, . . .. That is, at time point t+1, decided are u(t+1), . . . , u(t+m) usingthe manipulated variable time series deriving formula, with the newlymeasured y(t+1) and the previously delivered u(t) as known values, andamong them, u(t+1) is delivered to the thickness adjusting means 10.

The manipulated variable time series deriving step can be repeated everytime point as described above. Furthermore, it is also possible toderive manipulated variable time series, for example, at time points t,t+s, t+2s at s cycles, where s is an integer in a range of 2≦s≦m, and todeliver the u(t), . . . , u(t+s−1) derived at time point t in the periodfrom t to t+s−1.

The above-mentioned control action computation allows the sheetthickness values to be controlled to have a target thickness profilequickly and high accurately. That is, a process model, which formulatesthe interference phenomenon that if one thickness adjusting means ismanipulated, the sheet thickness values at the portions corresponding tothe adjacent adjusting means are changed, and which also formulates thedelay time and lag after manipulating one thickness adjusting means tillthe result appears in the measured thickness value at the correspondingposition, is used to decide a thickness prediction formula, and themanipulated variable time series for optimizing the formula are decidedand added. So, the sheet thickness values converge on the desired valuesvery quickly and very high accurately.

Furthermore, even if actually measured thickness values y becomedifferent from the prediction formula y_(P) as in the case where theprocess model contains an error or where any other disturbance occurs,quasi-optimum manipulated values can be decided through control withoutcausing the deviations due to the error or disturbance of the model tobe accumulated, if the decision of the prediction formula and thedecision of optimum manipulated variable time series are frequentlycarried out using newly measured thickness values y. So, the sheetthickness values can be controlled to approach the desired valuesquickly and very high accurately.

A parametric model is described above as the process model. However, forexample, the following models can also be used: an impulse responsemodel that describes how the sheet thickness values change at timepoints t=1, 2, 3, . . . , when impulse-wise outputs are given asmanipulated variables to the thickness adjusting means 10 at time pointt=0; a step response model that describes how the sheet thickness valueschange at time points t=1, 2, 3, . . . when step-wise outputs are givenas manipulated variables to the thickness adjusting means 10 at timepoint t=0; and a state space model that uses state variables to describethe relation between manipulated variables and state variables and therelation between state variables and sheet thickness values.

In the above-mentioned sheet production process, in the case where asheet is stretched in the transverse direction, the stretching conditioncan be regarded to be almost constant in the transverse direction in thecentral portion of the sheet, but as shown in FIG. 6 edge portions ofthe produced sheet are likely to be affected by the neck-in phenomenonthat the sheet width becomes more narrow immediately after the rawmaterial has been discharged from the die 4 than the width achievedwhile the raw material is discharged, or by the polymer flow at the dieedge portions. So, the sheet is stretched at the edge portions underconditions different from those in the central portion. Since the edgeportions are greatly affected by process conditions than the centralportion, in the case where the sheet thickness control means isdesigned, it is preferable that the process model is set differently inthe portions corresponding to the edge portions and the central portionin the transverse direction of the sheet as shown in FIG. 1. It is morepreferable to set the parameters participating in the decision of thecontrol means, for example, the process gain and interference ratedifferently for the respective portions.

A process gain refers to the rate of the controlled variable change tothe manipulated variable change. That is, it refers to a value how muchthe thickness of the sheet changes when a manipulated variable appliedto the sheet thickness adjusting means is changed by a unit quantity.

The borders between the central portion and the edge portions can alsobe decided in reference to the film forming conditions such as thestretching ratio in the transverse direction of the sheet and the sheetthickness and the distribution of sheet thickness values dispersed inthe transverse direction of the sheet. It is also preferable to changethe borders depending on the state of film production, for example, toset the width of the central portion at 70 to 80% and the width of theedge portions at the balance in the state of stable film production andto set the width at 60% or less when the sheet thickness in thetransverse direction is not stable as in the beginning of filmproduction.

As for the process gains, it is preferable to keep the process gains inthe edge portions smaller than the process gains in the central portion,since it can happen that in the edge portions of the sheet, the sheetthickness does not change so much as in the central portion as shown inthe thickness distribution of FIG. 6 due to the structural limit of thedie and the buffer effect of the polymer at the die edge portions, evenif the manipulated variables for the sheet thickness adjusting means arechanged by a certain quantity. That is, since the die may be fixed atthe outermost edge portions or since the several thickness adjustingmeans in the outermost edge portions may be fixed, it can happen thateven if manipulated variables are applied to the sheet thicknessadjusting means to keep the sheet thickness thinner in the edgeportions, the actually actuated strokes of the thickness adjusting meansbecome small, and furthermore, that since the polymer flow at the dieedge portions flow in, the actual sheet thickness in the edge portionsdoes not become as thin as that in the central portion. Similarly, evenif manipulated variables are applied to the sheet thickness adjustingmeans to make the sheet thickness thicker in the edge portions, it canhappen that the actually actuated strokes of the thickness adjustingmeans become small, or that since the polymer flows partially into thedie edge portions, the actual sheet thickness is in the edge portionsdoes not become as thick as that in the central portion.

As for the interference rates, it is preferable that the interferencerates at the positions corresponding to the adjacent adjusting means inthe edge portions are larger than in the central portion. The reasonsare the same as those described for the process gains. Since the die maybe fixed at the outermost edge portions or since several thicknessadjusting means in the outermost edge portions may be fixed, it canhappen that when manipulated variables are applied to the sheetthickness adjusting means in the edge portions, the differences from theactually actuated strokes at the positions of the adjacent thicknessadjusting means become small, and that since the polymer flow at the dieedge portions causes buffer action, the differences between the actualsheet thickness values in the manipulated positions and those in thepositions corresponding to the adjacent thickness adjusting means becomesmall.

Furthermore, for the above reasons, instead of setting the interferencerates in the edge portions symmetrically about a manipulated position,the interference rates on the edge portion side can be kept large whilethose on the central portion side are kept small.

Moreover, both the process gain and interference rate can be setrespectively differently for the right and left edge sides in the sheetrunning direction.

To decide the process gains and interference rates in the centralportion and the edge portions used for expressing the process model, itis preferable to measure the step responses of the thickness adjustingmeans. That is, if step-wise changing manipulated variables are appliedto thickness adjusting means representing the edge portions andthickness adjusting means representing the central portion, the processgains and interference rates in the edge portions and the centralportion can be measured from the changes in sheet thickness caused inthis case.

A method of designing the control means 9 based on the process gains andinterference rates decided according to the above methods is describedbelow.

When the control means is designed, the control parameter is decidedaccording to the method shown in FIG. 5. The control parameter to bedecided depends on the design method. At first, a process model isprepared based on the process gains, interference rates and delay timedecided according to the methods of the invention. Since the processgains in the edge portions of the film are smaller than those in thecentral portion, the control parameter is adjusted to keep the controlgains in the edge portions larger than those in the central portion, andthe process model is used to simulate the changes of sheet thickness. Ifthe control performance such as the control accuracy and theresponsiveness obtained by the simulation is better than the presetstandard values, the design parameter concerned is employed. If thecontrol performance is not good, the control parameter is re-adjusted tocarry out the simulation. This work is repeated to decide the optimumcontrol parameter.

If a process model prepared according to the invention is used tosimulate the changes of sheet thickness for deciding the controlparameter of the control means as described above, a sheet can beproduced efficiently since it can be avoided to decide the controlparameter by trial and error while forming the sheet.

If a process model suitable for the central portion and the edgeportions respectively is used for control as described above instead ofcontrolling the sheet uniformly entirely in the transverse direction ofthe sheet, the sheet thickness can be controlled accurately and stablyeven at the edge portions of the sheet that are likely to be affected bythe neck-in phenomenon and the polymer flow at the die edge portions,hence likely to be unstably controlled.

Meanwhile, since the die 4 is provided with integral lips in thetransverse direction of the sheet, the manipulated variable applied toeach thickness adjusting means 10 also affects the positions of the nearthickness adjusting means. So, in a state where the differences betweenthe manipulated variables of the respective near thickness adjustingmeans are excessively large, even if the differences between themanipulated variables of the respective near thickness adjusting meansare made larger, the form of the gap cannot follow the differences ofmanipulated variables. For this reason, the manipulated variables canless affect the changes of gap form of the die, and since the gapadjusting capability becomes low, the sheet thickness control accuracydeclines.

Therefore, in the sheet production process of the invention, in the casewhere the number of thickness adjusting numbers is N (N: a naturalnumber of 2 or more), it is preferable that in the case where thedifferences between the computed manipulated variable of the i-th (i=1,2, . . . , N) thickness adjusting means and the manipulated variablesdelivered to the thickness adjusting means near said means are notsmaller than a predetermined value T, the manipulated variable deliveredto said i-th thickness adjusting means is corrected to keep thedifferences of manipulated variables smaller, while the manipulatedvariables delivered to said near thickness adjusting means are correctedbased on a static process model expressing the static relation betweenmanipulated variables to be delivered and the sheet thickness values atthe position corresponding to the respective thickness adjusting means,to be obtained by the manipulated variables after lapse of a sufficienttime, and that the corrected respective manipulated variables aredelivered to said respective thickness adjusting means.

The control means 9 has a manipulated variable computing means 221,manipulated variable correcting means 222 and manipulated variabledelivering means 223 as shown in FIG. 8. The manipulated variablecomputing means 221 computes manipulated variables according to apredetermined control algorithm, and the manipulated variable correctingmeans 222 corrects manipulated variables when it is necessary to correctthem. The manipulated variable delivering means 223 delivers therespective corrected values actually to the thickness adjusting means10.

In the manipulated variable computing means 221, it is preferable toperform conversion processing such as filter processing for thedeviation data that are differential values between the thicknessdistribution of the sheet 1 and the desired thickness distribution. Forthe filter processing, such a method as the moving-averaging in the samedirection as the transverse direction of the sheet or theweighted-averaging with the deviation data obtained before the presenttime point can be used.

In most cases, the number of the thickness adjusting means 10 arrangedin the transverse direction of the sheet is smaller than the number ofsaid deviation data. In such a case, the data corresponding to therespective thickness adjusting means are taken from the filter-processeddeviation data. In this case, it is desirable to obtain beforehand thecorresponding positions between the respective thickness adjusting means10 and the deviation data.

Furthermore, the manipulated variable computing means 221 computes themanipulated variables for the filter-processed deviation data reduced toas many as the thickness adjusting means, for controlling the thicknessadjusting means 10.

In the case where the sheet thickness is controlled by theabove-mentioned method, it can happen that the manipulated variabledelivered to a thickness adjusting means is excessively large or smallcompared with the manipulated variables delivered to near thicknessadjusting means. FIG. 7 shows an example of manipulated variables 28delivered to respective thickness adjusting means 10. The position ofeach thickness adjusting means is selected as the abscissa, and themagnitude of the manipulated variable, as the ordinate. The magnitude ofthe manipulated variable can be, for example, the rate of time duringwhich a certain heat quantity is delivered to the thickness adjustingmeans concerned or the magnitude of the heat quantity delivered to thethickness adjusting means concerned during a predetermined period oftime. In the diagram, the manipulated variable of the thicknessadjusting means of position x or y is excessively larger or smaller thanthe manipulated variables a delivered to the thickness adjusting meansof the near positions x−1 and x+1, or y−1 and y+1.

The method of leveling the manipulated variables in this pattern ofmanipulated variables is described below.

If the manipulated variable 21 delivered to the i-th thickness adjustingmeans 10 _(i) (i=1, 2, . . . , N) is u_(i) and the measured thicknessvalue at the position corresponding to each thickness adjusting means 10_(i) at the moment when the manipulated variable is delivered is y_(i)(i=1, 2, . . . , N), then the static mathematical relation between u_(i)and y_(i) is, for example, as shown in formulae 19 and 20. The staticmathematical relation refers to the relation between sheet thicknessvalues and manipulated variables occurring after lapse of a sufficienttime subsequently to the application of the manipulated variables to thethickness adjusting means. In the formulae, A is an interference matrix,being a matrix with a size of N×N expressing the interferences betweenindividual thickness adjusting means. In formula 20, α₁ (≧0) is the rateat which the sheet thickness values at the positions corresponding toboth the first adjacent thickness measuring means change, and α₂ (≧0) isthe rate at which the sheet thickness values at the positionscorresponding to both the second adjacent thickness adjusting meanschange. In the above formula, the rate at which the sheet thicknessvalues at the positions corresponding to both the third and fartheradjacent thickness adjusting means change is assumed to be 0, but α₃(≧0) and rates of farther thickness adjusting means may also beconsidered. Furthermore, the values of α₁ and α₂ of respective rows canalso be different from row to row. $\begin{matrix}{{Formula}\quad 19\text{:}} \\{\begin{bmatrix}y_{1} \\y_{2} \\\vdots \\y_{N}\end{bmatrix} = {A\begin{bmatrix}u_{1} \\u_{2} \\\vdots \\u_{N}\end{bmatrix}}}\end{matrix}$ $\quad\begin{matrix}{{Formula}\quad 20\text{:}} \\{A = \begin{bmatrix}1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \cdots & \cdots & \cdots & \quad & 0 \\\alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & \quad & \quad & 0 \\\alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & \quad & 0 \\0 & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & 0 & \quad & \quad & 0 \\0 & 0 & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} & \alpha_{2} & ⋰ & \quad & \vdots \\0 & \quad & \quad & ⋰ & ⋰ & 1 & ⋰ & ⋰ & ⋰ & \vdots \\\vdots & \quad & \quad & \quad & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots \\\vdots & \quad & \quad & \quad & ⋰ & \quad & \quad & ⋰ & \quad & \alpha_{2} \\\vdots & \quad & \quad & \quad & \quad & ⋰ & \alpha_{2} & \alpha_{1} & 1 & \alpha_{1} \\0 & \quad & \cdots & \cdots & \cdots & \quad & 0 & \alpha_{2} & \alpha_{1} & 1\end{bmatrix}}\end{matrix}$

Considered is a case where the manipulated variables of the j-ththickness adjusting means 10 _(j) and respectively M thickness adjustingmeans on both sides, 10 _(j−M), 10 _(j−M+1), . . . , 10 _(j−1), 10_(j+1), . . . , 10 _(j+M−1) and 10 _(j+M) are corrected and leveled.

Assume that the correction rate of the manipulated variable delivered tothe thickness adjusting means 10 _(j) is a and that the correction ratesof the manipulated variables delivered to 2M thickness adjusting means10 _(j−M), 10 _(j−M+1), . . . , 10 _(j−1), 10 _(j+1), . . . , 10_(j+M−1) and 10 _(j+M) are b_(i)(i=−M, −M+1, . . . , −1, 1, . . . , M−1,M). The following describes, for simplification, a method of correctingthe manipulated variables delivered to respectively two thicknessmeasuring means on both sides of the thickness adjusting means 10 _(j)with M=2, but this method is applicable for cases other than M=2 andcases where the numbers of thickness adjusting means on both sides ofthe thickness adjusting means 10 _(j) are different. In these cases, theformulae shown below can be changed as required to obtain the correctionrates for leveling the manipulated variables.

If corrected manipulated variables are delivered to thickness adjustingmeans 10 _(j−2), 10 _(j−1), 10 _(j), 10 _(j+1), and 10 _(j+2), the sheetthickness values corresponding to the respective thickness adjustingmeans change by Δy_(i) (i=j−2, j−1, j, j+1, j+2). If it is assumed thateven if the corrected manipulated variables are applied to the thicknessadjusting means 10 _(j−2), 10 _(j−1), 10 _(j), 10 _(j+1) and 10 _(j+2),the sheet thickness values little change, the following formula holds.$\begin{matrix}{{Formula}\quad 21\text{:}} \\{\begin{bmatrix}{\Delta\quad y_{1}} \\{\Delta\quad y_{2}} \\{\Delta\quad y_{3}} \\{\Delta\quad y_{4}} \\{\Delta\quad y_{5}}\end{bmatrix} = {\begin{bmatrix}ɛ_{1} \\ɛ_{2} \\ɛ_{3} \\ɛ_{4} \\ɛ_{5}\end{bmatrix} = {A^{\prime}\begin{bmatrix}b_{1} \\b_{2} \\a \\b_{3} \\b_{4}\end{bmatrix}}}}\end{matrix}$In the above formula, A′ is the portions corresponding to the thicknessadjusting means 10 _(i) (i=j−2, j−1, j, j+1, j+2) to be corrected inmanipulated variable, extracted from A, and expresses the process modelnear the positions. The value of ε_(i) (i=1, 2, . . . , 5) is 0 or afinite value close to 0, preferably not more than 1.0% of the desiredsheet thickness.

As a method for obtaining b_(i) when a is given, it is preferable tosolve the following formulae 22 through 24 with ε_(i)=0 (i=1, 2, . . . ,5). Furthermore, a recurrence formula can be solved with ε_(i) (i=1, 2,. . . , 5) assumed to be infinite, or a recurrence formula can be solvedwith ε_(i)(i=1, 2, . . . , 5) assumed to be close to 0. $\begin{matrix}{{Formula}\quad 22\text{:}} \\{\begin{bmatrix}b_{1} \\b_{2} \\b_{3} \\b_{4}\end{bmatrix} = {a \cdot {A^{'' +}\begin{bmatrix}\alpha_{2} \\\alpha_{1} \\1 \\\alpha_{1} \\\alpha_{2}\end{bmatrix}}}}\end{matrix}$In the above formula, A″⁺ is a quasi inverse matrix of A″ and can beobtained from formula 23.Formula 23:A″ ⁺=(A″ ^(T) A″)⁻¹ A″ ^(T)where $\begin{matrix}{{Formula}\quad 24\text{:}} \\{A^{''} = \begin{bmatrix}1 & \alpha_{1} & 0 & 0 \\\alpha_{1} & 1 & \alpha_{2} & 0 \\\alpha_{2} & \alpha_{1} & \alpha_{1} & \alpha_{2} \\0 & \alpha_{2} & 1 & \alpha_{1} \\0 & 0 & \alpha_{1} & 1\end{bmatrix}}\end{matrix}$The above formulae 22 through 24 can be generalized into the followingformulae 25 and 26. $\begin{matrix}{{Formula}\quad 25\text{:}} \\{\begin{bmatrix}{\Delta\quad y_{- {M1}}} \\{\Delta\quad y_{{- {M1}} + 1}} \\\vdots \\{\Delta\quad y_{- 1}} \\{\Delta\quad y_{0}} \\{\Delta\quad y_{1}} \\\vdots \\{\Delta\quad y_{{M2} - 1}} \\{\Delta\quad y_{M2}}\end{bmatrix} = {\begin{bmatrix}ɛ_{- {M1}} \\ɛ_{{- {M1}} + 1} \\\vdots \\ɛ_{{M2} - 1} \\ɛ_{M2}\end{bmatrix} = {A^{\prime}\begin{bmatrix}b_{- {M1}} \\b_{{- {M1}} + 1} \\\vdots \\b_{- 1} \\a \\b_{1} \\\vdots \\b_{{M2} - 1} \\b_{M2}\end{bmatrix}}}} \\{{Formula}\quad 26\text{:}} \\{\begin{bmatrix}b_{- {M1}} \\b_{{- {M1}} + 1} \\\vdots \\b_{- 1} \\b_{1} \\\vdots \\b_{{M2} - 1} \\b_{M2}\end{bmatrix} = {a \cdot {A^{'' +}\begin{bmatrix}0 \\\vdots \\0 \\\alpha_{2} \\\alpha_{1} \\1 \\\alpha_{1} \\\alpha_{2} \\0 \\\vdots \\0\end{bmatrix}}}}\end{matrix}$A″⁺ is a quasi-inverse matrix of A″, and A″ is a matrix obtained byextracting the portion of (j+M2, j+M2) from the elements (j−M1, j−M1) ofmatrix A and excluding the j-th column of A.

If the above formulae are solved, the correction rates of themanipulated-variables can be derived and the manipulated variables canbe leveled with the sheet thickness little changed.

In the execution of leveling, it is preferable that the correction ratea of the manipulated variable applied to the j-th thickness adjustingmeans is not larger than 10% of the manipulated variable applied to thej-th thickness adjusting means. In the case where 10% or more ofcorrection is needed, it is preferable to carry out the leveling pluraltimes by several percent each. In this case, it is desirable that theleveling subsequent to the completion of one time of leveling is carriedout after the sheet thickness becomes stable.

As for the timing of leveling, after the control computation hasproduced the manipulated variables to be delivered to the respectivethickness adjusting means, the correction rates can be added to themanipulated variables. Alternatively the correction rates can beobtained after the manipulated variables have been applied to thethickness adjusting means before the next manipulated variables aredelivered, and added to the manipulated variables obtained in the nextcontrol computation, to deliver the corrected manipulated variables tothe thickness adjusting means.

An example of automatically leveling during thickness control isdescribed below based on FIG. 9. The following also describes, forsimplification, a method of correcting the manipulated variables of therespectively two thickness adjusting means on both sides of a thicknessadjusting means 10 with Ml M2=2 in formula 25.

At first, the control means 9 computes manipulated variables u_(k) (k=1,2, . . . , N) (step 1). Then, the difference D_(k) (=u_(k+1)−u_(k))between the manipulated variable u_(k) delivered to the thicknessadjusting means 10 _(k) at every position k and the manipulated variableu_(k+1) delivered to the thickness adjusting means 10 _(k+1) at positionk+1 is obtained (step 2).

In step 3, the respective absolute values of D_(k) and D_(k+1) arecompared with a predetermined threshold value, and the signs of D_(k)and D_(k−1) are examined.

Then, if the absolute values of D_(k) and D_(k+1) are larger than thepredetermined threshold value and D_(k) and D_(k+1) are different insign, that is, if the manipulated variable applied to the thicknessadjusting means of position k+1 is excessively larger or smaller thanthe manipulated variables applied to both the adjacent thicknessadjusting means, then a correction rate a, the magnitude of which is notlarger than 10% of the manipulated variable applied to the thicknessadjusting means 10 _(k−1) at position k+1, is applied to make thedifference smaller, and the correction rates b_(i) (i=k−1, k, k+2, k+3)delivered to the near thickness adjusting means 10 _(k−1), 10 _(k), 10_(k+1), 10 _(k+2) and 10 _(k+3) are obtained based on the formula 26(step 4). Then, the correction rates are applied to the correspondingmanipulated variables (step 5).

Subsequently, with the position changed from k to k=k+2M+1 (step 6),that is, for the thickness adjusting means adjacent to the positionapart enough not to doubly correct manipulated variables, steps 3through 5 are carried out.

The above steps are repeated till k≧N is reached (step 7), and thecorrected manipulated variables are actually delivered to the thicknessadjusting means (step 8).

Other methods of deciding the positions of the thickness adjustingmeans, the correction rates of which are obtained based on D_(k),include a method of deciding for a predetermined number of adjustingmeans selected in the descending order of absolute value of D_(k) and amethod of deciding based on the absolute value of the product ofadjacent differences D_(k) or the sum of the absolute values of adjacentdifferences D_(k), etc.

In the above method, the correction rate a can be set at a constant rateto the manipulated variable to be corrected, or can also be derived inresponse to the difference between the manipulated variables of adjacentthickness adjusting means, i.e., based on the magnitude of D_(i).

Furthermore, in addition to said a, b_(i) can also be given beforehandlike a, to derive other correction rates.

In the above method, leveling is carried out every control cycle, butcan also be carried out intermittently.

Another method for leveling a pattern of manipulated variables isdescribed below.

In the case where the number of thickness adjusting means is N (N: anatural number of 2 or more), it is also preferable to correct themanipulated variables delivered to consecutive M thickness adjustingmeans (M: a natural number of 2 to N) based on a process model thatexpresses the relation between the manipulated variables to be deliveredand the sheet thickness values to be obtained by the manipulatedvariables, in order to minimize the dispersion of the manipulatedvariables to be delivered to said consecutive M thickness adjustingmeans among the manipulated variables of the N thickness adjustingmeans.

If the manipulated variables to be delivered to the consecutive M (M≦N)thickness adjusting means 10 _(i) (i=1, 2, . . . , M) among the Nthickness adjusting means are u′_(i) and the sheet thickness changescorresponding to the respective thickness adjusting means 10 _(i) at themoment when the manipulated variables are delivered are y′_(i) (i=1, 2,. . . , M), then the process model can be expressed by Y′=B′U′. That is,the static mathematical relation between u′_(i) and y′_(i) is as shown,for example, by formulae 27 and 28. $\begin{matrix}{{Formula}\quad 27\text{:}} \\{\begin{bmatrix}y_{1}^{\prime} \\y_{2}^{\prime} \\\vdots \\y_{N}^{\prime}\end{bmatrix} = {A^{\prime}\begin{bmatrix}u_{1}^{\prime} \\u_{2}^{\prime} \\\vdots \\u_{N}^{\prime}\end{bmatrix}}}\end{matrix}$ $\begin{matrix}{{Formula}\quad 28\text{:}} \\{A^{\prime} = \begin{bmatrix}1 & \beta_{11} & \beta_{2} & 0 & \quad & \cdots & \cdots & \cdots & \quad & 0 \\\beta_{1} & 1 & \beta_{1} & \beta_{2} & 0 & \quad & \quad & \quad & \quad & 0 \\\beta_{2} & \beta_{1} & 1 & \beta_{1} & \beta_{2} & 0 & \quad & \quad & \quad & 0 \\0 & \beta_{2} & \beta_{1} & 1 & \beta_{1} & \beta_{2} & 0 & \quad & \quad & 0 \\0 & 0 & \beta_{2} & \beta_{1} & 1 & \beta_{1} & \beta_{2} & ⋰ & \quad & \vdots \\0 & \quad & \quad & ⋰ & ⋰ & 1 & ⋰ & ⋰ & ⋰ & \vdots \\\vdots & \quad & \quad & \quad & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots \\\vdots & \quad & \quad & \quad & ⋰ & \quad & \quad & ⋰ & \quad & \beta_{2} \\\vdots & \quad & \quad & \quad & \quad & ⋰ & \beta_{2} & \beta_{1} & 1 & \beta_{1} \\0 & \quad & \cdots & \cdots & \cdots & \quad & 0 & \beta_{2} & \beta_{1} & 1\end{bmatrix}}\end{matrix}$

In the above formulae, A′ is an interference matrix, being a matrix witha size of M×M expressing the interferences between individual thicknessadjusting means. In formula 28, β₁ (≧0) is the rate at which the sheetthickness values at the positions corresponding to both the firstadjacent thickness measuring means change when a manipulated variable ischanged at a certain thickness adjusting means, and β₂ (≧0) is the rateat which the sheet thickness values at the positions corresponding toboth the second adjacent thickness adjusting means change. In the aboveformula, the rate at which the sheet thickness values at the positionscorresponding to both the third and farther adjacent thickness adjustingmeans change is assumed to be 0, but β₃ (≧0) and the rates of fartherthickness adjusting means may also be considered. Furthermore, thevalues of β₁ and β₂ of respective rows can also be different from row torow.

A method of correcting the manipulated variables to be delivered to theconsecutive M thickness adjusting means for leveling the manipulatedvariables is described below based on FIG. 10.

The manipulated variables (u′₁, u′₂, . . . , u′_(M))^(T) not yetcorrected and to be delivered to the respective thickness adjustingmeans 10 _(j) are considered as vector U′.

At first, M pairs of eigenvalues λ_(i) (i=1, 2, . . . , M) andeigenvectors v_(i) (i=1, 2, . . . , M) of the interference matrix A′ areobtained (step 1). Furthermore, the magnitudes of the eigenvectors arenormalized to 1. In this case, A′ is a square matrix of M×M, and assumedis rank (A′)=M.

In this case, U can be expressed as follows, as the linear combinationof eigenvectors v_(i). $\begin{matrix}{{Formula}\quad 29\text{:}} \\{U^{\prime} = {\sum\limits_{i = 1}^{M}{a_{i}v_{i}}}}\end{matrix}$In the above formula, a_(i) is coefficients expressing the degrees atwhich eigenvectors v_(i) are contained in U′, and is obtained from thefollowing formula (step 2).Formula 30:(a ₁ , a ₂ . . . a _(M))^(T)=(v ₁ v ₂ . . . v _(M))⁻¹ U

In this case, the change of thickness y′_(i) by the change ofmanipulated variables u can be expressed as follows using v_(i), λ_(i)and a_(i). $\begin{matrix}{{Formula}\quad 31\text{:}} \\{Y^{\prime} = {\sum\limits_{i = 1}^{N}{a_{i}\lambda_{i}v_{i}}}}\end{matrix}$

Components relatively small in the magnitude of the product |a_(i)λ_(i)|of coefficient a_(i) and eigenvalue λ_(i), of the eigenvectors v_(i) arelarge in the influence on the dispersion of manipulated variables andsmall in the influence on the sheet thickness. So, even if a_(i)λ_(i)=0is assumed, the magnitude of sheet thickness Y′ little changes.

Then, |a_(i)λ_(i)| is evaluated with a preset threshold value T₁(0<T₁<1), and if the following equation is satisfied, assumed is a_(i)=0(steps 3 through 5).

Formula 32:|a _(i)λ_(i)|<max_(j)(|a _(j)λ_(j)|)·T₁,where max_(j)(|a_(j)λ_(j)|) is the maximum value of |a_(j)λ_(j)| (j=1,2, . . . , M).

The a_(i) (i=1, 2, . . . , M) corrected as described above is introducedinto the following formula, to derive corrected manipulated variablevector U=′″ (step 6). $\begin{matrix}{{Formula}\quad 33\text{:}} \\{U^{''} = {\sum\limits_{i = 1}^{N}{a_{i} \cdot v_{i}}}}\end{matrix}$

If the manipulated variables are corrected according to the aboveformula, the manipulated variables can be leveled with the sheetthickness little changed. If the threshold value T_(i) is made larger,the degree of leveling can be enhanced since many kinds of frequencycomponents can be removed from the manipulated variable vector U′, butthe influence on the sheet thickness tends to be larger. On the otherhand, if the threshold value T₁ is made smaller, the influence on thesheet thickness can be lessened, but the degree of leveling tends to belower. It is preferable that the threshold value T₁ is 0.01 to 0.5.

Furthermore, instead of changing the eigenvectors v_(i) to be removedfrom the manipulated variable vector U′ while depending on a_(i), themanipulated variables can also be corrected using said formula 33,assuming that the a_(i) corresponding to the eigenvalue λ_(i) satisfyingthe following formula is 0.

Formula 34:|λ_(i)|<max_(j)(|λ_(j)|)·T ₂where 0<T₂<1, preferably 0.01≦T₂≦0.5.

Finally, the corrected manipulated variables are delivered to thethickness adjusting means (step 7).

In the invention, the leveling can be carried out every control cycle,but can also be carried out intermittently. Furthermore, carrying outthe leveling in the case where a certain condition is satisfied is alsoa preferable mode. For example, it is preferable to carry out levelingin the case where the variance of the manipulated variables delivered toconsecutive M thickness adjusting means has become larger than a presetvalue, or in the case where the difference between the maximum value andthe minimum value of the manipulated variables delivered to M thicknessadjusting means becomes larger than a preset value, or in the case wherethe difference between the manipulated variables delivered to adjacentthickness adjusting means becomes larger than a preset value.

In the invention, the number M of the thickness adjusting means to beleveled is 2 to N. A case of M=N, i.e., where M is equal to the numberof all the thickness adjusting means (N), is preferable since themanipulated variables of all the thickness adjusting means can beleveled as a package. M can also be set at any desired values smallerthan N. In this case, since computation covers only the portions thatmust be leveled, computation can be simplified.

Another preferable embodiment in which the invention is applied to theproduction of a plastic film is described below.

The thickness adjusting means 10 can be any of heat bolt method, lipheater method, etc. In the heat bolt method, heat quantities are givento heat bolts, to change their temperatures, thereby thermally expandingor contracting them, for adjusting the gap 11 of the die 4. In the lipheater method, lip heaters are used to change the temperatures of thepolymer, to change the viscosities of the polymer so that the amounts ofthe polymer extruded from the die 4 can be changed to adjust thethickness values of the sheet 1.

As described above, in the heat bolt method and the lip heater method,the temperature is adjusted to adjust the sheet thickness.

In the heat bolt method, if the temperature of each thickness adjustingmeans 10 is raised, the sheet thickness becomes thinner since the gap ofthe die is narrowed. On the contrary, in the lip heater method, if thetemperature is raised, the sheet thickness becomes thicker since theviscosity of the polymer declines to increase the amount extruded fromthe die.

In the method of controlling the sheet thickness values by controllingthe temperatures of heating type thickness adjusting means, when thetemperatures are raised, the thickness adjusting means 10 are forciblyheated, for example, by applying electric powers as described above, butwhen the temperatures are lowered, the thickness adjusting means 10 arenaturally allowed to cool in most cases.

In the above-mentioned method, the change of the sheet thickness withthe lapse of time when the temperature is raised is different from thatwhen the temperature is lowered. Compared with the case of raising thetemperature, the change of the sheet thickness with the lapse of timeoccurs slowly when the temperature is lowered.

So, in the case where heating type thickness adjusting means are usedand where the manipulated variables are controlled by controlling theheat quantities applied to the heating type thickness adjusting means,it is preferable that the variations of the heat quantities are madelarger when the heat quantities are decreased than when they areincreased.

In the control means 9 of this embodiment, as shown in FIG. 4, themanipulated variable computing means 21 computes the heat quantities 23based on the differential values 25 between the measured thicknessvalues 24 and the desired thickness values of the sheet 1, and thecomputed heat quantities are applied to the thickness adjusting means 10of the sheet production process 26. In many cases, since electricheaters are used, the heat quantities are given as electric currentsapplied to the heating type thickness adjusting means.

The control means 9 computes the heat quantities based on thefilter-processed deviation data reduced to as many as the number of thethickness adjusting means, to control the thickness adjusting means 10.

The changes of sheet thickness caused when the temperatures of heatbolts or lip heaters are changed, and the method of computing the heatquantities in the invention are described below.

As described before, in the heat bolt method, if the temperatures of thethickness adjusting means 10 are raised, the sheet thickness valuesbecome smaller since the gap of the die become smaller. On the contrary,in the case of viscosity method, if the temperatures are raised, thesheet thickness values become larger since the viscosities of thepolymer are lowered to increase the amounts extruded from the die.

The changes of sheet thickness with the lapse of time in the case ofheating the thickness adjusting means 10 and in the case of cooling themin the heat bolt method are described below in reference to FIG. 12. Toheat the thickness adjusting means 10, the heat quantities applied tothe thickness adjusting means are increased. On the contrary, in thecase of cooling the thickness adjusting means 10, the heat quantitiesapplied to the thickness adjusting means are decreased.

FIG. 12 is a schematic diagram showing the change 31 of sheet thicknesswith the lapse of time in the case of heating a heat bolt and the change30 of sheet thickness with the lapse of time in the case of cooling theheat bolt, with the, same heat quantity variation given in reversedirections. It can be seen that the change of sheet thickness in thecase of cooling the heat bolt is slow compared with that in the case ofheating the heat bolt.

Furthermore, FIG. 13 shows a schematic diagram showing the changes ofsheet thickness with the lapse of time with the heat bolt cooled with alarge heat quantity and a small heat quantity.

It can be seen that the sheet thickness change 35 in the case of a largeheat quantity is faster than the sheet thickness change 36 in the caseof a small heat quantity.

FIG. 13 also shows the sheet thickness change 37 with the lapse of timein the case where the heat quantity is small and where the time constantof the process is small i.e. the sheet thickness changes quickly. When acertain time 34 has elapsed after a heat quantity is applied to thethickness adjusting means 10, the sheet thickness variation is the sameas that in the case where the heat quantity is large.

Therefore, if the certain time 34 is a control cycle, it can beconsidered that the same sheet thickness change as that in the casewhere the time constant is small occurs in this control cycle if theheat quantity is made larger. That is, as shown in FIG. 11(a), if thebasic heat quantity computing means 211 computes the variation of heatquantity (basic heat quantity) to be given on the assumption that theresponsiveness during cooling is the same as that during heating indefiance of actually asymmetric responsiveness, and the heat quantityfor control is obtained to ensure that the variation becomes γtimes(γ>1), then the responsiveness in the case of cooling the heat boltcan be improved. In this case, the heat quantity in the case of coolinga heat bolt is made larger in reference to the case of heating the heatbolt, but on the contrary, the heat quantity in the case of heating theheat bolt can be made smaller in reference to the case of cooling theheat bolt (first mode).

Furthermore in FIG. 12, if the sheet thickness variation 33 in the caseof heating the bolt occurring when a certain time 34, has elapsed afterthe application of a heat quantity to the thickness adjusting means 10is compared with the sheet thickness variation 32 in the case of coolingthe bolt, it can be seen that the sheet thickness variation 33 in thecase of heating the bolt is larger.

So, in the case where a heat quantity is obtained based on thedifferential value between a measured thickness value and the desiredvalue and where a control method of making the variation of heatquantity larger when the differential value is larger is employed, asshown in FIG. 11(b), if the control differential value computing means213 converts the differential value during cooling to ensure that thedifferential value during cooling becomes α times (α>1) that duringheating, for making the differential value for control beforehand, andsubsequently the heat quantity computing means 214 computes the heatquantity as ordinary control computation, then the variation of heatquantity becomes large, and the responsiveness in the case of coolingthe bolt can be enhanced as in the first mode (second mode).

Also in this case, on the contrary, in the case of heating, the controldifferential value between said measured value and said desired valueused for cooling can be multiplied by β (0<β<1), to obtain the heatquantity and it can be applied to the heating type thickness adjustingmeans (third mode).

It is desirable that the above α, β and γ are decided considering theprocess gains, time constant of heating/cooling and control cycles.Furthermore, α, β and γ can also be variable depending on the magnitudesof the differences between desired thickness values and measuredthickness values, instead of being constant values.

In the above example, added is a means of making asymmetric, the inputor output of a control system that performs ordinary symmetriccomputation. However, the control system can be constituted to allow thecomputation of such asymmetric control output.

A further other preferable embodiment in which the invention is appliedto the production of a plastic film is described below.

In the equipment of producing a sheet such as a film, the sheet is woundaround a winder, and slight roughness of the sheet in the transversedirection can cause a winding mound as shown in FIG. 15.

In the prior art, to prevent the occurrence of such a winding mound, anoscillator is installed upstream of the winder, to oscillate the sheetin the transverse direction, i.e., crosswise while the sheet is wound.If the sheet is oscillated crosswise to move thick portions crosswise inthe transverse direction like this, the occurrence of a winding moundcan be prevented.

If the sheet is wound while being moved crosswise as described above,the wound sheet cannot be uniform at the edges, and the winding looksawkward. In addition, there is a problem that the process into thesecondary product is inconvenienced.

So, it is preferable to measure the thickness distribution of the sheetin the transverse direction, to obtain the integral values of thedifferences between the measured thickness distribution values and thefirst target values preset based on the desired outer diameter profileof the produced roll, to correct the second target values at respectiveportions in the transverse direction based on the integral values, andto control the manipulated variables applied to the thickness adjustingmeans using said evaluation function to lessen the differences betweenthe second target values and the measured sheet thickness distributionvalues.

The action of the control means 9 is described below in detail inreference to FIG. 14. At first, an integrating circuit 41 integrates thedeviations 44 between the measured thickness values 24 of the sheetformed according to the sheet production process measured by thethickness gauge 8 at the respective measuring positions after a givenmeasuring start point, and the first target values 42 at the positionscorresponding to the measured thickness values.

It is desirable that the values per sheet obtained by converting thedesired outer diameter profile in the transverse direction of theproduced roll are used as the first target values 42. The outer diameterprofile can be, for example, flat or drum-shaped in the transversedirection, and any profile desired by the user can be set.

Then, a second target value correcting means 40 corrects the secondtarget values at the positions corresponding to the integral valuesbased on the integral values. As the second target values, for example,the desired thickness values of one sheet can be set. For correction,the second target values can be corrected at rates proportional to themagnitudes of the integral values, or PID control or the like can beused. In this case, it is desirable that an upper limit value and alower limit value are preset for the second target values, to preventthat the second target values are corrected beyond the upper or lowerlimit value.

The manipulated variable computing means 21 derives the manipulatedvariables for the thickness adjusting means 10 in order to lessendeviations between said corrected second desired values 43 and themeasured thickness values 24 using said evaluation function. However, itis preferable to filter-process the deviation data 25 beforehand. Forthe filter processing, such a method as the moving-averaging in the samedirection as the transverse direction of the sheet or theweighted-averaging with the deviation data obtained before the presenttime point can be used. It is desirable to decide the manipulatedvariables depending on the situations in the production process, forexample, to correct large thickness irregularity immediately after startof sheet production and to correct small thickness irregularity duringstable production.

Furthermore, in most cases, the number of the thickness adjusting means10 disposed in the transverse direction of the sheet is smaller than thenumber of measured deviation data. In such a case, the datacorresponding to the respective thickness adjusting means are taken fromthe filter-processed deviation data. The data can be obtained using afunction of thickness distribution and the fitting of the least squaremethod, etc. When the number of measured data is sufficiently large, thecorresponding data can also be obtained by simply thinning out. In thiscase, it is desirable to obtain the corresponding relation between therespective thickness adjusting means 10 and the measured thicknessvalues 24 beforehand.

Then, the manipulated variable computing means 21 computes themanipulated variables 23 based on the filter-processed deviation datareduced to as many as the number of the thickness adjusting means, andthe manipulated variable delivering means 22 delivers the manipulatedvariables 23 to the thickness adjusting means 10 of the sheet productionequipment 26.

In the above method, all the number of the first desired values 42, thenumber of the second desired values 43, the number of measured thicknessvalues 24, and the number of deviation data 44 and 25 are equal to thenumber of the thickness values measured by the thickness gauge 8, andthe manipulated variable computing means 21 takes the values of thepositions corresponding to the respective thickness adjusting means 10and computes the manipulated variables based on these values. However,the first desired values 42, the second desired values 43, the measuredthickness values 24 and the deviation data 44 and 25 can also be reducedto only the data of the positions corresponding to the respectivethickness adjusting means 10 before hand. Furthermore, since thecorrection of the second desired values is not necessarily carried outfrequently, when the integrating means 4 integrates the differences fromthe first desired values, it is not necessary to use the latest valuesat that moment. Therefore, the values measured at an adequately lessfrequency can be used, or the values measured by any other means thanthat for the measured values used in the manipulated variable computingmeans 21 can also be used. Moreover, the correction of second desiredvalues can also be carried out at longer cycles.

Furthermore, depending on the characteristics peculiar to the process,it can happen that the first desired values are different from the outerdiameter profile of the produced roll. In such a case, if the outerdiameter profile in the transverse direction of the produced roll ismeasured and the first desired values are corrected based on themeasured values, then the outer diameter profile of the produced rollcan be improved. The outer diameter profile can be measured using astylus method or laser displacement meter, etc. Since the windingdiameter is small when little time has passed after start of winding,the velocity of increasing the winding diameter is wound more than whena certain time has passed after start of winding. So, when the windingdiameter is small, the thickness irregularity affects the profile of theproduced roll greatly. Therefore, if the correction rates of the seconddesired values are made smaller as the winding diameter of the sheetbecomes larger, the sheet can be produced without being affected by thewinding diameter.

Furthermore, in the sheet production process, there is a case whereafter a roll with a length as long as integer times a certain length isproduced and slit, and each of the slit rolls is rewound into pluralrolls each having said certain length. In this case, it is preferablethat the profiles of all the produced rolls are good. Therefore, in thecase where plural rolls each having a certain length are rewound fromone produced roll, it is preferable to integrate and reset for everysaid certain length, and to newly obtain the integral values of thedifferences between the measured values and preset first desired values.The winding length till resetting is not required to be always constant,and the resetting can be carried out whenever any adequately necessarypredetermined value has been reached.

One of the causes to worsen the winding appearance of a roll is thethickness irregularity of the sheet. However, the conventional thicknessirregularity decreasing technique as described above has variousproblems. The inventors studied intensively and as a result found thatif a specific component of sheet thickness irregularity is selectivelydecreased, a slightly wrinkled or streaked roll with a good roll formcan be obtained without changing the properties of the sheet with highproductivity sustained. Particularly, the thickness profile in thetransverse direction of the sheet is measured and Fourier-transformedinto a power spectrum resolved into wave numbers, and the sheetthickness is controlled to ensure that the mean value X1 of the powersof smaller than a predetermined wave number a becomes 0.2×T² or less andnot larger than the mean value X2 of the powers of wave number a andmore in said power spectrum. According to a study of the inventors, theinfluence of the component of smaller than a specific wave length on thewinding appearance is large, and if the mean value X1 of the powers ofsmaller than said wave number a is 0.2×T² or less, the irregularity ofthe winding appearance of the roll formed by winding the sheet can bekept very small. In this case, T is the mean thickness (μm) of thesheet. It is preferable that X1 is 0.1×T² or less. Furthermore, if themean value X2 of the powers of said wave number a and larger is largerthan the mean value X1 of said powers, high productivity can besustained without changing the properties of the sheet. A range ofX1≦0.5×X2 is preferable, and a range of X1≦0.2×X2 is more preferable.

The predetermined wave number a can be set depending on the kind of thesheet and production conditions, and it is preferable that the wavenumber is any optional value selected in a range from 3 m⁻¹ to 30 m⁻¹.The low frequency component of smaller than this wave number worsens thewinding appearance of the roll since the thickness irregularity isaccumulated. On the other hand, the high frequency component of largerthan this wave number does not affect the winding appearance of the rollso much.

The invention is described below in reference to drawings based on anembodiment where the invention is applied to the production of a plasticfilm.

The method of measuring the thickness profile of a plastic film(hereinafter simply called the film) in the transverse direction is asdescribed below. For example, several sheets to tens of sheets are cutout from optional winding layers of a produced roll, and the thicknessesof discretely sampled portions can be measured, for example, using acontact measuring instrument. It is preferable that the samplingintervals are 1 mm or less. Furthermore, the measurement can also becarried out at different positions in the longitudinal direction of thefilm, to average the power spectra obtained in the respective times ofmeasurement.

The obtained thickness profile of the sheet in the transverse directionare Fourier-transformed according to the following formula, to obtainthe power spectrum P of the respective wave numbers. P = F(ω)(F(ω))^(*)F(ω) = ∫_(−∞)^(∞)f(x)𝕖^(−jω  x)𝕕xwhere f(x) is the thickness profile of the sheet in the transversedirection (in μm); F(ω) is the Fourier transform of f(x); x is aposition in the transverse direction of the sheet (in μm); ω (in m⁻¹) isa wave number; and F(ω)* is a conjugate complex number; and j is animaginary number, and j²=−1.

The mean value X1 of the powers of a predetermined wave number a andlarger and the mean value X2 of the powers of smaller than the wavenumber a are obtained as described below. That is, X1 is the mean valueof the powers whose wave number is more than 0 and smaller than the wavenumber a, and shows the degree to which the thickness irregularity ofthe low frequency component is contained in the film. X2 is the meanvalue of the powers of the predetermined wave number a and larger, andshows the degree to which the thickness irregularity of the highfrequency component is contained in the film. The upper limit of thewave number computed when X2 is obtained can be the upper limit of theobtained power spectrum. However, if too high wave numbers are includedin the computation, noise may be contained. So, it is preferable that X2is the mean value of the powers of the predetermined wave number a to100 m⁻¹, and it is more preferable that X2 is the mean value of thepowers of the predetermined wave number a to 40 m⁻¹.

It is preferable that the predetermined number a is equal to 1/(aninterval between respectively adjacent thickness adjusting means x thestretching ratio of the sheet in the transverse direction). Thethickness irregularity of wave numbers of larger than it little affectsthe winding appearance of the roll and the control by the thicknessadjusting means becomes difficult.

If the film is oscillated when wound, the influence of thicknessirregularity can be further decreased. The oscillation means toreciprocate the roll as a wound film in the transverse direction of theroll when the film is wound. If the film is oscillated especially in anamplitude range of 0.5×(1/wave number a)≦(Oscillationamplitude)≦5×(1/wave number a), when wound, the thickness irregularityof high wave numbers little affects the winding appearance.

If the sheet production process of the invention using theabove-mentioned control action computation is used, the sheet thicknesscan be controlled to have a desired thickness profile quickly at highaccuracy. Thus, the low frequency component in the thicknessirregularity of the sheet can be efficiently removed.

The respective actions of the control means in the above embodiments canbe realized by means of a computer, a program for causing those actions,etc. The program and the data of various storing means can bedistributed by tangible media available for computer reading such asfloppy disc, MO and CD-ROM and transmission means such as wired orwireless network.

Examples in which the invention is used to produce sheets are describedbelow.

EXAMPLE 1

The sheet production equipment shown in FIG. 2 was used to produce a 2.7μm thick polyester film. The width of the produced film was 3.5 m, andthe film forming speed in the product portion was 175 m/min. Each of thethickness adjusting means 10 was a heat bolt containing a cartridgeheater for thermally expanding and contracting the bolt, therebyadjusting the gap 11. The number of heat bolts used for thicknesscontrol was 45. The thickness gauge 8 used was a light interference typethickness gauge using the interference phenomenon of light described inJP, 4-522, B. The thickness gauge can measure the film thickness at 15mm intervals in the transverse direction of the film while scanning atcycles of 60 seconds in the transverse direction of the film. Thecontrol interval were 60 seconds equal to the scanning cycles of thethickness gauge.

The process model of formula 1 was decided as shown by the followingformula, based on the sheet thickness changes near the measuringposition corresponding to one heat bolt to which a predeterminedmanipulated variable was applied In this example, a manipulated variablerefers to the rate of time during which a heat quantity is applied toeach heat bolt. ${{Equation}\quad 35{\text{:}\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\\vdots \\\vdots \\\vdots \\\vdots \\y_{n - 2} \\y_{n - 1} \\y_{n}\end{bmatrix}}} = {\frac{{- 0.315_{z}^{- 2}} - 0.012_{z}^{- 1} + 1.57}{0.012_{z}^{- 3} + 0.54_{z}^{- 2} - 0.714_{z}^{- 1}}{\begin{matrix}\begin{bmatrix}0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65 & 0.25 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.25 & 0.65 & 1 & 0.65\end{bmatrix}\end{matrix}\begin{bmatrix}u_{1} \\u_{2} \\u_{3} \\\vdots \\\vdots \\\vdots \\\vdots \\\vdots \\\vdots \\u_{m - 2} \\u_{m - 1} \\u_{m}\end{bmatrix}}}$

Furthermore, the L, P and m of formula 10 were 0, 10 and 7 respectively,and coefficients λ_(i) (i=1, 2, . . . , P) and Ψ_(i) (i=1, 2, . . . , m)of the evaluation function J were set at 1.0 and 0.8 respectively.

At first, manipulated variables were applied to plural heat bolts tointentionally make thickness irregularity (obtained by dividing thedifference between the maximum value and the minimum value of thicknessby a mean thickness value), and the thickness of the film was controlledaccording to the method of the invention.

Furthermore, for comparison, thickness irregularity of a similar samedegree was made, and the thickness of the film was controlled accordingto conventional control (PID control).

FIG. 16 shows the control result of PID control, and FIG. 17, thecontrol result of the method of the invention. In FIG. 16, in control ofabout 30 minutes, the thickness irregularity was improved from 8.4% toonly 7.4%, but in FIG. 17, similarly in control of about 30 minutes, thethickness irregularity was improved from 9.1% to 1.4%. It could beconfirmed that when the method of the invention was used, the sheetthickness could be controlled to have a desired profile quickly at goodaccuracy.

EXAMPLE 2

Thickness irregularity of a similar degree to that of Example 1 wasmade. Immediately after start of control when the thickness irregularitywas still high, coefficients λ_(i) (i=1, 2, . . . , P) and Ψ_(i) (i=1,2, . . . , m) of the evaluation function J were set at 1.0 and 0.5respectively to keep the contribution of Ψ relating to the manipulatedvariables small, for applying large manipulated variables, and when thethickness irregularity became 5%, the respective coefficients werechanged to 1.0 and 0.8 respectively. As a result, the thicknessirregularity could be improved faster than in Example 1.

EXAMPLE 3

The range of respectively 35% on both sides of a center in thetransverse direction of a film was defined as the central portion, andthe remaining ranges of 15% each on both sides, as the edge portions.Predetermined manipulated variables were applied to respectivelyoptional heat bolts in the central portion and the edge portions. Whenthe sheet thickness values near the sheet thickness measuring positionscorresponding to the respective heat bolts became stable, the sheetthickness distribution was referred to for deciding the α₁ and α₂ asα₁=0.6 and α₂=0.2 in the central portion and α₁=0.7 and α₂=0.3 in theedge portions. Thickness irregularity of a similar degree to those ofExamples 1 and 2 was made in the entire width of the sheet, and themethod of the invention was applied to control the film thickness. As aresult, the thickness became more uniform over the entire width of thefilm than in Examples 1 and 2, and the film could be formed stably.

EXAMPLE 4

The sheet production equipment shown in FIG. 2 described above was usedto produce a 2.7 μm thick polyester film. Each of the thicknessadjusting means 10 was a heat bolt containing a cartridge heater forthermally expanding and contracting the bolt, thereby adjusting the gap11. The number of heat bolts used for thickness control was 45, and therespective bolts were arranged at a pitch of 20 mm. The thickness gauge8 used was a light interference type thickness gauge using theinterference phenomenon of light. The width of the produced film was 3.5m, and the film forming speed in the product portion was 175 m/min.

In this example, to make a process model at first according to thecontrol means design flow of FIG. 5, the manipulated variables ofoptional heat bolts were changed step-wise to measure the changes ofthickness, as step response test. The step response test was carried outin the edge portions and the central portions to measure the changes ofthickness. As a result, it was found that:

-   (1) The process gains were smaller in the edge portions than in the    central portion.-   (2) The interference rates were different in the edge portions and    the central portion.-   (3) It was suitable that the central portion accounted for 70% of    the entire width while the edge portions accounted for 15% each.

The measured values were as shown in Table 1.

TABLE 1 Central portion Edge portions Process gain 0.05 μm/% 0.03 μm/%Interference rate Controlled position 1.0 1.0 First adjacent positions0.6 0.7 Second adjacent positions 0.2 0.3 Delay time Td 80 sec Timeconstant T 160 sec

In the above, a process gain refers to the rate of the sheet thicknesschanges to the manipulated variable changes applied to a heat bolt. Themanipulated variable of a heat bolt is the percentage of the time duringwhich a certain electric power is supplied to the heat bolt, to acertain period time (10 seconds). Therefore, the unit of a process gainis μm/%.

An interference rate expresses the change of the sheet thickness at thepositions correspond to a certain operated heat bolt, and a firstadjacent position or second adjacent position is of the bolt caused whensaid heat bolt is operated, with the change at the position correspondto the operated heat bolt as 1.

Based on the results of the step response test, in this example, aprocess model different in two process parameters of process gain andinterference rate between the edge portions and the central portion wasprepared.

That is, the static mathematical relation between the thickness changeYi at the position corresponding to each heat bolt and the manipulatedvariable change Ui applied to the heat bolt in the central portion isexpressed by the following formulaYi=0.05‡(0.2Ui−2+0.6Ui−1+Ui+0.6Ui+1+0.2Ui+2)  (1)and that in the edge portions, by the-following formulaYi=0.03‡(0.3Ui−2+0.7Ui−1+Ui+0.7Ui+1+0.3Ui+2)  (2)where

-   Ui: Manipulated variable change of i-th position heat bolt [%]-   Yi: Thickness change at the corresponding position [μm]

Furthermore, it was assumed that time constant T and dead time Tdexisted as dynamic characteristics of the process model. The timeconstant T and the delay time Td were identified from the step responsetest. In this example, as shown in Table 1, equal values were obtainedfor the edge portions and the central portion.

FIG. 18 shows the control result of sheet thickness obtained when thethickness of the plastic film produced using the controller decidedbased on the process model described above was stabilized after lapse ofsufficient time. The control range was from No. 11 bolt to No. 52 bolt(corresponding to the positions of 600 mm and 310 mm from the edges ofthe produced film), and the central portion ranged from No. 18 bolt toNo. 45 bolt. As shown in the diagram, the thickness became uniform overthe entire width, and the film could be formed stably.

On the other hand, FIG. 19 shows the control result of sheet thicknessobtained when the plastic film was produced using the controller decidedbased on a process model havingYi=0.05‡(0.2Ui−2+0.6Ui−1+Ui+0.6Ui+1+0.2Ui+2)  (3)in the entire width. While the actual process gains in the edge portionswere small, the gains of the control system were adapted to the centralportion. So, the entire gain became insufficient, and the thicknesscontrol stability was poor. Even after lapse of sufficient time,thickness irregularity remained.

EXAMPLE 5

An example in which the invention was used to produce a plastic film isdescribed below.

The production equipment shown in FIGS. 2 and 3 with a die 4 having 40heat bolts used as the thickness adjusting means 10 was used to producea plastic film.

A manipulated variable in this example refers to the rate of time duringwhich a heat quantity is applied to a heat bolt, per predeterminedperiod of time.

For components of the interference matrix of formula 20, the filmproduction process was analyzed to set α₁=0.75 and α₂=0.35.

The leveling was carried out automatically as shown in FIG. 9. Thethreshold value T in step 3 of FIG. 9 was set at 15%, and the correctionrate a was set at −5%. The thickness adjusting means corrected at a timewere one thickness adjusting means greatly different in the manipulatedvariable from both the adjacent thickness adjusting means, and therespectively two adjacent thickness adjusting means on both sides of it,five thickness adjusting means in total.

With ε_(i)=(i=1, 2, . . . , 5) in formula 21, a pseudo-inverse matrixwas used for leveling as shown in formulae 22 through 24.

D_(i) was obtained for each manipulated variable, and when it satisfied|D_(i)|>T=15, |D_(i−1)|>T=15 and D_(i)×D_(i−1)<0, the correction rate ofthe manipulated variable was obtained according to formula 22.

The pattern of manipulated variables at a certain time point during thesheet thickness control was as shown in FIG. 20, and the pattern of thedifferential values between the manipulated variables delivered toadjacent thickness adjusting means was as shown in FIG. 21. In FIGS. 20and 21, the magnitude of the manipulated variable is chosen as theordinate of this graph, and it expresses the rate of the time duringwhich a certain heat quantity is delivered to a heat bolt, perpredetermined period of time. The value of the manipulated variable atposition i+1 projects by about 20% compared with the values of both theadjacent positions. In FIG. 21, |D_(i)| and |D_(i+1)| are larger thanthe threshold value and different in sign. So the manipulated variable40 (=u_(i+1)) was leveled. As a result, in formula 22, the correctionrates b_(i) became b₁=b₄=−1.0% and b₂=b₃=3.0% respectively.

The calculated correction rates were added to the manipulated variables,and the sums were delivered to the thickness adjusting means.

FIG. 22 shows the change with the lapse of time, of the measuredthickness value corresponding to the thickness adjusting means atposition i+1 that received the manipulated variable 60. The sheetthickness little changed with the lapse of time.

For comparison, the manipulated variables were corrected to reduce thedifferences between the manipulated variable 60 and the manipulatedvariables of both the adjacent positions to 10% or less in the patternof manipulated variables as shown in FIG. using the method described inJP, 7-329147, A Gazette. The change of the sheet thickness with thelapse of time in this case is shown in FIG. 23. The sheet thicknesschanged greatly to impair the quality of the sheet.

EXAMPLE 6

An example in which the invention was used to produce a plastic film isdescribed below.

The production equipment shown in FIGS. 2 and 3 with a die 4 having 38heat bolts used as the thickness adjusting means 10 was used to producea plastic film.

A manipulated variable in this example refers to the rate of the timeduring which a heat quantity was applied to a heat bolt, perpredetermined period of time.

For components of the interference matrix of formula 28, the filmproduction process was analyzed to set β₁=0.75 and β₂=0.35.

It was decided to carry out the leveling of manipulated variables whenthe standard deviation of the manipulated variables delivered to the 38thickness adjusting means was 3% or more. With M=38 and T₁=0.05 informula 1, the manipulated variables of all the 38 thickness adjustingmeans were leveled to be corrected.

The pattern of manipulated variables at a certain time point during thesheet thickness control became as shown in FIG. 24. The magnitude of themanipulated variable is chosen as the ordinate of this graph, and itexpresses the rate of the time during which a certain heat quantity wasdelivered to a heat bolt, per predetermined period of time. Since thestandard deviation of the manipulated variables at this moment was 3.5%,the manipulated variables were leveled. FIG. 25 shows the leveledmanipulated variable pattern applied to the thickness adjusting means.As a result, the standard deviation of manipulated variables became2.3%. So, the leveling could make the standard deviation of manipulatedvariables smaller. FIG. 26 shows the sheet thickness distribution beforeleveling, and FIG. 27, the sheet thickness distribution after leveling.The thickness distribution after leveling was little different from thatbefore leveling.

For comparison, manipulated variables were corrected sequentially in thedescending order of the magnitude of the difference between themanipulated variables of respectively adjacent thickness adjustingmeans, in order to become the difference between the manipulatedvariables at both the adjacent positions 7% or less in the pattern ofmanipulated variables as shown in FIG. 24 using the method described inJP07329147A. The sheet thickness distribution obtained when the standarddeviation of manipulated variables became 2.3% is shown in FIG. 28. Thesheet thickness distribution greatly changed.

EXAMPLE 7

An example in which the invention was used to produce a plastic film isdescribed below.

In a film production process using heat bolts, certain heat quantitieswere applied to plural bolts, to push or pull the bolts. The mean values71 and 73 in the change of film thickness with the lapse of time in thiscase are shown in FIGS. 29 and 30. In the diagrams, broken lines 72 and74 are obtained by functionally approximating 71 and 73. Comparing thetime taken for the approximation curve 72 to change from 10% of theentire variation to 90% (dotted lines 75 and 76 in FIG. 29) with thatfor the approximation curve 76 to change similarly (dotted lines 75 and76 in FIG. 30), the time in the case of heating (pushing) (72) isshorter than that in the case of cooling (pulling) (74) by a factor ofabout 1.4. FIG. 31 shows the outer diameter profile 81 of the rollproduced in this case. Since the pulling action was slow andinsufficient, the outer diameter profile was generally thinner than thedesired value.

So the variation of a heat quantity was increased to 1.4 times in thecase of cooling, to produce a film as described above. FIG. 31 shows theouter diameter profile 82 of the roll produced in this case. When largerheat quantities were applied in the case of cooling bolts, theresponsiveness during cooling improved, and the irregularity of theouter diameter profile of the produced roll generally decreased.

In the case where the measured values are smaller than the desired valuewith heat bolts used, the corresponding heat bolts are cooled. In such acase, if the differences between measured values and the desired valueare increased to 1.4 times, to obtain heat quantities, the irregularityof the outer diameter profile of the roll generally decrease as in thisexample.

EXAMPLE 8

An example in which the invention was used to produce a plastic film isdescribed below.

The sheet production equipment shown in FIG. 2 as described above wasused to produce a 2.7 μm thick polyester film. As the thicknessadjusting means 10, heat bolts each containing a cartridge heater forthermally expanded and contracted to adjust the gap 11 were used. Thenumber of the heat bolts used for thickness control was 45, and thebolts were arranged at a pitch of 20 mm. The sheet width of the productportion was 3.5 m, and the stretching ratio in the transverse directionof the sheet was 3.5 times.

For computation control of manipulated variables, a step of measuringthe thickness distribution of the sheet in the transverse direction, astep of deriving manipulated variable time series in which apredetermined evaluation function for evaluating the future sheetthickness changes predicted based on said measured values and on aprocess model expressing the relation between said manipulated variablesand sheet thickness values becomes a minimum value, and a step ofdelivering at least the first manipulated variables of said manipulatedvariable time series to said thickness adjusting means were repeated atpredetermined intervals.

The wound roll was divided into three rolls, and from the respectivesurface layers of the rolls, 10 sheets each were cut out, and thethickness of the sheets were measured by a contact film thickness gaugeat 0.5 mm intervals. The power spectrum 90 was as shown in FIG. 32.

With wave number a at 1/(an interval between thickness adjusting means xstretching ratio of the sheet in the transverse direction)=0.014, themean value X1 of the powers of smaller than wave number a and the meanvalue X2 of the powers of wave number a and larger were found to beX1=25 and X2=120. X1 was sufficiently small compared with X2. The rollform of the roll of the film was good.

In the above examples, heat bolts were used as the thickness adjustingmeans, but the thickness adjusting means are not especially limited ifthey suit the objects of the invention. For example, actuators such asservomotors or pneumatic motors and die bolts can also be used. Theelements for changing the manipulated variables of thickness adjustingmeans can be other than energizing times and voltages, such as therotating angles of die bolts and heater temperatures.

INDUSTRIAL APPLICABILITY

As described above, since the method of manufacturing sheet of theinvention can control the sheet thickness to have a desired thicknessprofile at high speed, the time taken after starting the production ofthe sheet till a predetermined thickness irregularity level acceptableas a product is reached can be shortened to remarkably reduce thequantity of non-conforming products produced in this duration and toenhance the production efficiency, thus allowing the sheet cost to bereduced. Furthermore, even if the thickness profile of the sheetchanges, for example, since the temperature distribution of thestretching machine changes during film formation, it can be controlledto have a desired profile quickly. So, as a result, the thicknessuniformity of the sheet can be improved to improve the quality of thesheet.

1. A method of manufacturing a sheet, in which a raw material isextruded and molded into a sheet using a die with plural thicknessadjusting means and the thickness of said sheet is controlled bycontrolling the manipulated variables applied to said thicknessadjusting means, characterized by repeating, at predetermined intervals,a step of measuring the thickness distribution of the sheet in thetransverse direction, a step of deriving manipulated variable timeseries in which a predetermined evaluation function for evaluating thefuture sheet thickness changes predicted based on said measured valuesand on a process model expressing the relation between said manipulatedvariables and sheet thickness values becomes a minimum value, and a stepof delivering at least the first manipulated variables of the derivedmanipulated variable time series to said thickness adjusting means.
 2. Amethod of manufacturing a sheet according to claim 1, wherein saidpredetermined evaluation function is based on said sheet thicknesschanges and said changes of manipulated variables.
 3. A method ofmanufacturing-a sheet according to claim 2, wherein as saidpredetermined evaluation function, different evaluation functions areused, in the beginning of production and during stable production, toensure that the contribution of sheet thickness changes becomes higherrelatively to the contribution of manipulated variables in the beginningof production than during stable production.
 4. A method ofmanufacturing a sheet according to claim 1, wherein said process modelused is expressed by a product obtained by multiplying a transferfunction and a constant matrix in which at least the diagonal componentis not zero.
 5. A method of manufacturing a sheet according to claim 4,wherein constants respectively different in the portions correspondingto the edge portions and the central portion of the sheet in thetransverse direction are used as said constant matrix.
 6. A method ofmanufacturing a sheet according to claim 1, wherein the number of thethickness adjusting means is N (N: a natural number of 2 or more); themanipulated variable delivered to the i-th (i=1, 2, . . . , N) thicknessadjusting means is corrected to lessen the differences between thederived manipulated variable of the i-th thickness adjusting means andthe manipulated variables delivered to the thickness adjusting meansnear the i-th thickness adjusting means in the case where the saiddifferences are not less than a predetermined value, T, while themanipulated variables delivered to said near thickness adjusting meansare corrected based on a static process model expressing the staticrelation between the manipulated variables to be delivered and the sheetthickness values, to be obtained by them after lapse of a sufficienttime; and the respectively corrected manipulated variables are deliveredto said respective thickness adjusting means.
 7. A method ofmanufacturing a sheet according to claim 6, wherein when the correctionrate of the manipulated variable to be delivered to said i-th thicknessadjusting means is a, the correction rates b₁ (j=−M1, −M1+1, . . . , −1,1, . . . , M2−1, M2) of the manipulated variables to be delivered to theM1 and M2 (M1, M2: natural numbers) thickness adjusting meansrespectively adjacent to said i-th thickness adjusting means on bothsides are derived using the following formula: $\begin{bmatrix}ɛ_{- {M1}} \\ɛ_{{- {M1}} + 1} \\\vdots \\ɛ_{{M2} - 1} \\ɛ_{M2}\end{bmatrix} = {A^{\prime}\begin{bmatrix}b_{- {M1}} \\b_{{- {M1}} + 1} \\\vdots \\b_{- 1} \\a \\b_{1} \\\vdots \\b_{{M2} - 1} \\b_{M2}\end{bmatrix}}$ where if A′ is a matrix with a size of(M1+M2+1)×(M1+M2+1) obtained by extracting the portions corresponding tothe i-th thickness adjusting means and the (M1+M2) thickness adjustingmeans respectively consecutively positioned adjacent to the i-ththickness adjusting means on both sides, from a matrix with a size ofN×N ,in case that the static process model is presented, as Y=AU, whereU is the vector of the manipulated variables (u₁, u₂, . . . , u_(N))delivered to N thickness adjusting means while Y is the vector of the Nsheet thickness values (y₁, y₂, . . . , y_(N)) corresponding to therespective thickness adjusting means; and ε_(i) (i=−M1, −M1*1, . . . ,M2−, M2) is 0 or a finite value.
 8. A method of manufacturing a sheetaccording to claim 1, wherein the number of thickness adjusting means isN (N: a natural number of 2 or more), and the manipulated variables tobe delivered to consecutive M (M: a natural number of 2 to N) thicknessadjusting means are corrected based on a static process model expressingthe static relation between the manipulated variables to be deliveredand the sheet thickness values at the position corresponding to therespective thickness adjusting means, to be obtained by them after lapseof a sufficient time, to lessen the dispersion of the manipulatedvariables delivered to said consecutive M thickness adjusting meansamong the manipulated variables of said N thickness adjusting means. 9.A method of manufacturing a sheet according to claim 8, wherein if themanipulated variables (u′₁, u′₂, . . . , u′_(M))^(T) delivered to saidconsecutive M thickness adjusting means are vector U′, the M sheetthickness values (y′₁, y′₂, . . . , y′_(M))^(T) corresponding to therespective thickness adjusting means are vector Y′, and said processmodel is expressed by Y′=A′U′ (A′ is a matrix with a size of M×M), thenthe corrected manipulated variable vector U″=(u₁″, u₂″, . . . , u_(M)″)are derived from the following formula, using the coefficients a_(i)(i=1, 2, . . . , M) of M eigenvectors v_(i) (i=1, 2, . . . , M) ofmatrix A′ obtained by resolving the manipulated variable vector U′before correction by said eigenvectors, the eigenvalues λ_(i) (i=1, 2, .. . , M) corresponding to said eigenvectors, and predetermined thresholdvalue T₁ (0<T₁<1) respectively:$U^{''} = {\sum\limits_{i = 1}^{N}{{f_{i}\left( {a_{i},\lambda_{i}} \right)} \cdot v_{i}}}$$f_{i} = \left\{ \begin{matrix}{a_{i},} & \left( {{{a_{i}\lambda_{i}}} \geq {\max_{j}{\left( {{a_{j}\lambda_{j}}} \right) \cdot T_{1}}}} \right) \\{0,} & \left( {{{a_{i}\lambda_{i}}} < {\max_{j}{\left( {{a_{j}\lambda_{j}}} \right) \cdot T_{1}}}} \right)\end{matrix} \right.$
 10. A method of manufacturing a sheet according toclaim 8, herein if the manipulated variables (u′₁, u′₂, . . . ,u′_(M))^(T) delivered to said consecutive M thickness adjusting meansare vector U′, the M sheet thickness values (y′₁, y′₂, . . . ,y′_(M))^(T) corresponding to the respective thickness adjusting meansare vector Y′, and said static process model is represented by Y′=A′U′(A′ is a matrix with a size of M×M), then the corrected manipulatedvariable vector U′″=(u₁″, U₂″, . . . , u_(M)″) are derived from thefollowing formula, using the coefficients a_(i) (i=1, 2, . . . , M) of Meigenvectors v_(i) (i=1, 2, . . . , M) of matrix A′ obtained byresolving the manipulated variable vector U′ before correction by saideigenvectors, the eigenvalues λ_(i) (i=1, 2, . . . , M) corresponding tosaid eigenvectors, and predetermined threshold value T₂ (0<T₂<1)respectively.$U^{''} = {\sum\limits_{i = 1}^{N}{{f_{i}\left( {a_{i},\lambda_{i}} \right)} \cdot v_{i}}}$$f_{i} = \left\{ \begin{matrix}{a_{i},} & \left( {{\lambda_{i}} \geq {\max_{j}{\left( {\lambda_{j}} \right) \cdot T_{2}}}} \right) \\{0,} & \left( {{\lambda_{i}} < {\max_{j}{\left( {\lambda_{j}} \right) \cdot T_{2}}}} \right)\end{matrix} \right.$
 11. A method of manufacturing a sheet according toclaim 1, wherein the thickness adjusting means are heating typethickness adjusting means; the control of manipulated variables is tocontrol the heat quantities applied to said heating type thicknessadjusting means; and the variations of heat quantities are made largerwhen the heat quantities are decreased than when the heat quantities areincreased.
 12. A method of manufacturing a sheet according to claim 1,wherein the thickness distribution of the sheet in the transversedirection is measured; the integral values of the differences betweensaid measured thickness distribution values and the first target presetbased on the desired outer diameter profile of the produced roll areobtained; the second target in the respective portions in the transversedirection are corrected based on said integral values; and themanipulated variables applied to said thickness adjusting means arecontrolled based on the differences between said second target and themeasured sheet thickness distribution values using said evaluationfunction.
 13. A device for controlling sheet thickness, which deliversmanipulated variables obtained based on the measured sheet thicknessvalues at respective portions of a sheet in the transverse directionmeasured by a thickness gauge to sheet thickness adjusting means atcorresponding positions; comprising a manipulated variable time seriesderiving means for deriving manipulated variable time series in which apredetermined evaluation function for evaluating the future sheetthickness changes predicted based on said measured values and on aprocess model expressing the relation between said manipulated variablesand sheet thickness values becomes a minimum value and a manipulatedvariable delivering means for delivering at least the first manipulatedvariables of the derived manipulated variable time series to saidthickness adjusting means.
 14. A device for controlling sheet thicknessaccording to claim 13, wherein said manipulated variable time seriesderiving means uses different evaluation functions in the beginning ofproduction and during stable production, as said predeterminedevaluation function, to ensure that the contribution of sheet thicknesschanges become higher relatively to the contribution of manipulatedvariables in the beginning of production than during stable production.15. A device for controlling sheet thickness according to claim 13,wherein said process model used is expressed by a product obtained bymultiplying a transfer function and a constant matrix in which at leastthe diagonal component is not zero.
 16. A device for controlling sheetthickness according to claim 13, wherein said manipulated variable timeseries deriving means uses constants different respectively in theportions corresponding to the edge portions and the central portions ofthe sheet in the transverse direction, as said constant matrix.
 17. Adevice for controlling sheet thickness according to claim 13, whereinthe number of the thickness adjusting means is N (N: a natural number of2 or more); and provided are a manipulated variable computing means forcomputing the manipulated variables applied to the respective adjustingmeans, and a manipulated variable correcting means for correcting themanipulated variable delivered to the i-th (i=1, 2, . . . , N) thicknessadjusting means, to lessen the differences between the computedmanipulated variable of the i-th thickness adjusting means and themanipulated variables delivered to the thickness adjusting means nearthe i-th thickness adjusting means in the case where the saiddifferences are not less than a predetermined value, T, while correctingthe manipulated variables delivered to said near thickness adjustingmeans based on a static process model expressing the relation betweenthe manipulated variables to be delivered and the sheet thickness valuesat the position corresponding to the respective thickness adjustingmeans, to be obtained by them after lapse of a sufficient time.
 18. Adevice for controlling sheet thickness according to claim 13, whereinthe number of the thickness adjusting means is N (N: a natural number of2 or more); and provided is a manipulated variable correcting means forcorrecting the manipulated variables delivered to consecutive M (M: anatural number of 2 to N) thickness adjusting means based on a staticprocess model expressing the relation between the manipulated variablesto be delivered and the sheet thickness values at the positioncorresponding to the respective thickness adjusting means, to beobtained by them after lapse of a sufficient time, to lessen thedispersion of the manipulated variables delivered to the saidconsecutive M thickness adjusting means among the manipulated variablesof N thickness adjusting means.
 19. A device for controlling sheetthickness according to claim 13, wherein provided are an integratingmeans for obtaining the integral values of the differences between themeasured thickness values of the respective portions of the sheet in thetransverse direction and the first target preset based on a desiredouter diameter profile of the produced roll, a second target correctingmeans for correcting the second target at the respective portions in thetransverse direction based on the values obtained by the integratingmeans, and a manipulated variable computing means for computing themanipulated variables to be applied to the thickness adjusting meansbased on the differences between the values obtained by the seconddesired value correcting means and the measured sheet thickness values.20. A program, for letting a computer perform the action of repeating,at predetermined intervals, a step of entering the measured thicknessvalues at the respective portions of a sheet in the transversedirection, a step of computing the differences between the targetthickness values and the measured thickness values at the respectiveportions, and a step of computing the manipulated variables to beapplied to thickness adjusting means based on the said differences atthe respective portions, characterized in that the step of computingmanipulated variables includes a step of deriving manipulated variabletime series in which a predetermined evaluation function for evaluatingthe future sheet thickness changes predicted on said measured values andon a process model expressing the relation between said manipulatedvariables and sheet thickness values becomes a minimum value, and a stepof delivering at least the first manipulated variables of the derivedmanipulated variable time series to said thickness adjusting means. 21.A program according to claim 20, wherein when the future sheet thicknesschanges obtained based on said measured sheet thickness values and on aprocess model expressing the relation between said manipulated variablesand the sheet thickness values are evaluated using a predeterminedevaluation function, used is a process model in which the relationbetween the manipulated variables of the thickness adjusting means andthe sheet thickness values in the edge portions in the transversedirection of the sheet is different from the corresponding relation inthe central portion.
 22. A program according to claim 20, wherein thenumber of thickness adjusting means is N (N: a natural number of 2 ormore); and the step of computing manipulated variables includes a stepof correcting the manipulated variable to be delivered to the i-th (i=1,2, . . . , N) thickness adjusting means, to lessen the differencesbetween the computed manipulated variable of the i-th thicknessadjusting means and the manipulated variables to be delivered to thenear thickness adjusting means in the case where the said differencesare not less than a predetermined value, T, while correcting themanipulated variables to be delivered to said near thickness adjustingmeans based on a static process model expressing the relation betweenthe manipulated variables to be delivered and the sheet thickness valuesat the position corresponding to the respective thickness adjustingmeans, to be obtained by them after lapse of a sufficient time, anddelivering the corrected respective manipulated variables to therespective thickness adjusting means.
 23. A program according to claim20, wherein the number of thickness adjusting means is N (N: a naturalnumber of 2 or more); and the step of computing manipulated variablesincludes a step of correcting manipulated variables based on a staticprocess model expressing the relation between the manipulated variablesto be delivered and the sheet thickness values at the positioncorresponding to the respective thickness adjusting means, to beobtained by them after lapse of a sufficient time, to lessen thedispersion of the manipulated variables delivered to consecutive M (M: anatural number of 2 or more) thickness adjusting means among thecomputed manipulated variables, and delivering the corrected respectivemanipulated variables to the respective thickness adjusting means.
 24. Aprogram according to claim 20, wherein the step of computing manipulatedvariables includes a step of deriving the integral values of thedifferences between the measured thickness distribution values and thefirst target preset based on the desired outer diameter profile of theproduced roll, a step of correcting the second target at the respectivemeasuring positions based on the said integral values, a step ofcomputing the differences between the target thickness values and saidmeasured values at the respective portions, and a step of computing themanipulated variables applied to the sheet thickness adjusting meansbased on the differences between the second target values and themeasured sheet thickness values.
 25. A program according to claim 20,wherein the thickness adjusting means are heating type thicknessadjusting means; the control of manipulated variables controls the heatquantities applied to the heating type thickness adjusting means; andthe step of computing manipulated variables computes the heat quantitiesto ensure that the heat quantity variations become larger when the heatquantities are decreased than when they are increased.
 26. A storagemedium capable of being read by a computer, that stores the program asset forth in claim
 20. 27. A sheet obtained by extruding and molding araw material using a die with plural thickness adjusting means,characterized in that the power spectrum of the thickness profile of thesheet in the transverse direction expressed by the following formulaP = F(ω)(F(ω))^(*) F(ω) = ∫_(−∞)^(∞)f(x)𝕖^(−jω  x)𝕕x (where f(x) is thethickness profile of the sheet in the transverse direction (in μm), F(ω)is the Fourier transform of f(x), x is a position in the transversedirection of the sheet (in m), ω is a wave number (in m⁻¹), and F(ω)* isthe conjugate complex number of F(ω); and j is an imaginary number, andj²=−1) and the mean sheet thickness T (μm) satisfy the followingrelation: The mean value X1 of the powers of smaller than apredetermined wave number a is 0.2×T² or less and is smaller than themean value X2 of the powers of wave number a and larger.
 28. A sheetaccording to claim 27, wherein in the power spectrum of the thicknessprofile of the sheet in the transverse direction, the mean value X1 ofthe powers of smaller than a predetermined wave number a is smaller thanthe mean value X2 of the powers of wave number a to 100 m⁻¹.
 29. A sheetaccording to claim 27, that satisfies X1≦0.5×X2.
 30. A sheet accordingto claim 27, wherein the wave number a is 3 m⁻¹ to 30 m⁻¹.
 31. A sheetaccording to claim 27, wherein the wave number a is equal to 1/(aninterval between thickness adjusting means×stretching ratio of the sheetin the transverse direction).
 32. A sheet according to claim 27, thatsatisfies 0.5×(1/wave number a)≦(amplitude of oscillation)≦5×(1/wavenumber a).
 33. A sheet according to claim 27, which is a plastic film.